Download On Advanced Analytic Number Theory

On Advanced Analytic Number Theory
By
C.L. Siegel
Tata Institute of Fundamental Research
Mumbai, India
i
FOREWORD
“Advanced Analytic Number Theory” was first published by the Tata Institute of Fundamental Research in their Lecture Notes series in 1961. It is now
being made available in book form with an appendix–an English translation
of Siegel’s paper “Berechnung von Zetafunktionen an ganzzahligen Stellen”
which appeared in the Nachrichten der Akademie der Wissenschaften in Göttingen,
Math-Phy. Klasse. (1969), pp. 87-102.
We are thankful to Professor Siegel and to the Göttingen Academy for
according us permission to translate and publish this important paper. We
also thank Professor S. Raghavan, who originally wrote the notes of Professor Siegel’s lectures, for making available a translation of Siegel’s paper.
K. G. Ramanathan
ii
PREFACE
During the winter semester 1959/60, I delivered at the Tata Institute of
Fundamental Research a series of lectures on Analytic Number Theory. It was
may aim to introduce my hearers to some of the important and beautiful ideas
which were developed by L. Kronecker and E. Hecke.
Mr. S. Raghavan was very careful in taking the notes of these lectures and
in preparing the manuscript. I thank him for his help.
Carl Ludwig Siegel
Contents
1 Kronecker’s Limit Formulas
1
The first limit formula . . . . . . . . . .
2
The Dedekind η-function . . . . . . . .
3
The second limit formula of Kronecker .
4
The elliptic theta-function ϑ1 (w, z) . . .
5
The Epstein Zeta-function . . . . . . .
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1
1
13
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29
41
2 Applications of Kronecker’s Limit Formulas to Algebraic Number
Theory
52
1
Kronecker’s solution of Pell’s equation . . . .√. . . . . . . . . 52
2
Class number of the absolute class field of Q( d)(d < 0) . . . 69
3
The Kronecker Limit Formula for real quadratic fields and its
applications . . . . . . √. . . . . . . . . . . . . . . . . . . . . 75
4
Ray class fields over Q( √d), d < 0 . . . . . . . . . . . . . . . 89
5
Ray class fields over Q( D), D > 0. . . . . . . . . . . . . . . 102
6
Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . 133
3 Modular Functions and Algebraic Number Theory
1
Abelian functions and complex multiplications . .
2
Fundamental domain for the Hilbert modular group
3
Hilbert modular functions . . . . . . . . . . . . . .
1
Elliptic Modular Forms . . . . . . . . . . . . . . .
2
Modular Forms of Hecke . . . . . . . . . . . . . .
3
Examples . . . . . . . . . . . . . . . . . . . . . .
iii
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149
149
165
184
220
225
231
Chapter 1
Kronecker’s Limit Formulas
1 The first limit formula
1
Let s = σ + it be a complex variable, σ and t being real. The Riemann zetafunction ζ(s) is defined for σ > 1 by
ζ(s) =
∞
X
k−s ,
k=1
here k−s stands for e−s log k , with the real value of log k. The series converges
absolutely for σ > 1 and uniformly in every s-half-plane defined by σ ≥ 1 +
ǫ(ǫ > 0). It follows from a theorem of Weierstrass that the sum-function ζ(s)
is a regular function of s for σ > 1. Riemann proved that ζ(s) possesses
an analytic continuation into the whole s-plane which is regular except for a
simple pole at s = 1 and satisfies the well-known functional equation
!
s
1−s
−((1−s)/2)
−s/2
ζ(1 − s),
ζ(s) = π
Γ
π Γ
2
2
Γ(s) being Euler’s gamma-function.
We shall now prove
Proposition 1. The function ζ(s) can be continued analytically into the halfplane σ > 0 and the continuation is regular for σ > 0, except for a simple pole
at s = 1 with residue 1. Further, at s = 1, ζ(s) has the expansion
ζ(s) −
1
= C + a1 (s − 1) + a2 (s − 1)2 + · · ·
s−1
1
Kronecker’s Limit Formulas
2
C being Euler’s constant.
We first need to prove the simplest form of a summation formula due to
Euler, namely.
Lemma. If f (x) is a complex-valued function having a continuous derivative
f ′ (x) in the interval 1 ≤ x ≤ n, then
!
Z 1 X
 n−1 ′
f (1) + f (n)
1 

f (x + k) x −  dx +
2
2
0
k=1
=
n
X
k=1
f (k) −
Z
n
f (x)dx.
(1)
1
2
Proof. In face, if a complex-valued function g(x) defined in the interval 0 ≤
x ≤ 1, has a continuous derivative g′ (x), then we have from integration by
parts,
!
Z 1
Z 1
1
1
′
g (x) x =
dx = (g(0) + g(1)) −
g(x)dx.
(2)
2
2
0
0
(Formula (2) has a simple geometric interpretation in that the right hand side of
(2) represents the area of the portion of the (x, y)-plane bounded by the curve
y = g(x) and the straight lines y − g(0) = (g(1) − g(0))x, x = g(0) and x = (g1)).
In (2), we now set g(x) = f (x + k), successively for k = 1, 2, . . . , n − 1. We then
have for k = 1, 2, . . . , n − 1,
!
Z 1
Z k+1
1
1
f ′ (x + k) x −
dx = ( f (k) + f (k + 1)) −
f (x)dx.
2
2
0
k
Adding up, we obtain formula (1).
This is Euler’s summation formula in its simplest form, with the remainder
term involving only the first derivative of f (x).
It is interesting to notice that if one uses the fact that the Fourier series
P e2πinx
(n = 0, omitted) converges uniformly to x − 1/2 in any closed in−
n 2πin
terval (ǫ, 1 − ǫ)(0 < ǫ < 1) one obtains from (2) the following useful result,
namely,
If f (x) is periodic in x with period 1 and has a continuous derivative, then
Z 1
∞
X
e2πinx
f (x)e−2πinx dx.
f (x) =
n=−∞
0
Kronecker’s Limit Formulas
3
Proof of Proposition 1. Let us now specialize f (x) to be the function f (x) =
x−s = e−x log x (log x taking real values for x > 0). The functionx−s is evidently
continuously differentiable in the interval 1 ≤ x ≤ n; and applying (1) to the
function x−s , we get
−s
=
n−1 Z
X
k=1
n
X
k=1
1
(x + k)
−s−1
0
k−s −
Z
!
1
1
x−
dx + (1 + n−s )
2
2
n
x−s dx
1
P
1−s
n


(if s , 1)
 k=1 k−s − 1−n
s−1
=
P

 nk=1 k−1 − log n(if s = 1).
(3)
Let us now observe that the right-hand side of (3) is an entire function of s.
Rn
We suppose now that σ > 1. When n tends to infinity 1 x−s dx tends to
n
P
k−s converges to ζ(s). Thus for
1/(s − 1). Further, as n tends to infinity,
k=1
σ > 1, the right hand side of (3) converges to ζ(s) − 1/(s − 1) as n tends to
infinity.
On the other hand, let the left-hand side of (3) be denoted by ϕn (s). Then
ϕn (s) is an entire funciton of s and further, for σ ≥ ǫ > 0,
|ϕn (s)| ≤
≤
n
1 X −1−ǫ 1 + n−ǫ
k
+
|s|
2 k=1
2
∞
1 X −1−ǫ
k
+ 1.
|s|
2 k=1
Thus, as n tends to infinity, ϕn (s) converges to a regular function of s in the
half-plane σ > 0; this provides the analytic continuation of ζ(s) − 1/(s − 1) for
σ > 0.
Now the constant a0 in the power-series expansion at s = 1 of
ζ(s) −
1
= a0 + a1 (s − 1) + a2 (s − 1)2 + · · ·
s−1
is nothing but lim ϕn (1). In other words,
n→∞
a0 lim 1 +
n→∞
!
1
1
+ · · · + − log n
2
n
3
Kronecker’s Limit Formulas
4
= C (Euler’s constant).
Proposition 1 is thus proved.
The constant C lies between 0 and 1. It is not known whether it is rational
or irrational; very probably, it is irrational.
One could determine the constants a1 , a2 , . . . also explicitly but this is more
complicated.
We shall consider now an analogous problem leading to the Kronecker
limit formula (Kroneckersche Grenzformle).
Instead of the simple linear function x, we consider a positive-definite binary quadratic form Q(u, v) = au2 + 2buv + cv2 in the real variables u and v
and with real coefficients a, b, c (we then have a > 0 and ac − b2 = d > 0).
Associated with Q(u, v), let us define
ζQ (s) =
∞
X
(Q(m, n))−s ,
4
(4)
m,n=−∞
P
where ′ denotes summation over all pairs of integers (m, n), except (0, 0).
Q(u, v) being positive-definite, there exists a real number λ > 0, such that
Q(u, v) ≥ λ(u2 + v2 ) and it is an immediate consequence that the series (4)
converges absolutely for σ > 1 and uniformly in every half-plane defined by
σ ≥ 1 + ǫ(ǫ > 0). Thus ζQ (s) is a regular function of s, for σ > 1.
As in the case of ζ(s) above, we shall obtain an analytic continuation of
ζQ (s) regular in the half-plane σ > 1/2, except for a simple pole at s = 1. The
constant a0 in the expansion at s = 1 of ζQ (s), viz.
ζQ (s) =
a−1
+ a0 + a1 (s − 1) + · · ·
s−1
is given precisely by the Kronecker limit formula. The constant a−1 which is
the residue of ζQ (s) at s = 1 was found by Dirichlet in his investigations on the
class-number of quadratic fields.
Letus first effect some simplifications.
Since Q(u, v) = Q(−u, −v), we see that
ζQ (s) = 2
∞
X
(Q(m, 0))−s + 2
m=1
Further
!2
!
b
b2 2
Q(u, v) = a u + v + c −
v
a
a
∞
∞ X
X
n=1 m=−∞
(Q(m, n))−s .
(5)
5
Kronecker’s Limit Formulas
5
√
√



!2

b
d 2
b − −d 
b + −d  
= a u + v + v = a u +
v u +
v
a
a
a
a
= a(u + zv)(u + zv)
√
√
where arg( −d) = π/2 and z = (b + −d)/a = x + iy with y > 0.
Also, we could, without loss of generality, suppose that d = 1; if Q1 (u, v) =
√
(1/ d)(au2 + 2buv + cv2 ) = a1 u2 + 2b1 uv + c1 v2 , then ζQ1 (s) = d s/2 ζQ (s) and
a1 c1 − b21 = 1. The function d s/2 (choosing a fixed branch) is a simple entire
function of s and therefore, to study ζQ (s), it is enough to consider ζQ1 (s).
Then we have a = y−1 and Q(u, v) = y−1 (u + zv)(u + zv) = y−1 |u + zv|2 . Now (5)
becomes
ζQ (s) = 2y s
∞
X
m−2s + 2y s
∞
∞ X
X
n=1 m=−∞
m=1
= 2y s ζ(2s) + 2y s
∞
∞ X
X
n=1 m=−∞
|m + nz|−2s
|m + nz|−2s
(6)
We know from Proposition 1 that ζ(2s) has an analytic continuation in the
half-plane σ > 0, regular except for a simple pole at s = 1/2. To obtain
an analytic continuation for ζQ (s), we have, therefore, only to investigate the
nature of the second term on the right hand side of (6), as a function of s. For
this purpose, we need the Poisson summation formula.
Proposition 2. Let f (x) be continuous in (−∞, ∞) and let
f (x + m) =
m=−∞
Proof. The function ϕ(x) =
∞
X
−2πikx
e
k=−∞
Z∞
f (x + m) be
m=−∞
uniformly convergent in 0 ≤ x ≤ 1. Then
∞
X
∞
P
f (ξ)e2πikξ dξ.
−∞
6
∞
P
f (x + m) is continuous in (−∞, ∞) and peR1
riodic in x, of period 1. If ak = 0 ϕ(ξ)e2πikξ dξ then, by Féjér’s Theorem, the
∞
∞
P
P
|ak | converges,
ak e−2πikx is equal to ϕ(x). In particular, if
(C, 1) sum of
m=−∞
k=−∞
k=−∞
then for x in (−∞, ∞)
∞
X
m=−∞
f (x + m) = ϕ(x) =
∞
X
k=−∞
e−2πikx
Z
0
1
ϕ(ξ)e2πikξ dξ.
Kronecker’s Limit Formulas
6
∞
P
In view of the uniform convergence of
f (x + m) in 0 ≤ x ≤ 1,
m=−∞
Z
1
∞
X
∞ Z
X
f (ξ + m)e2πikξ dξ =
0 m=−∞
m=−∞
∞
Z
=
Thus
∞
X
f (x + m) =
m=−∞
∞
X
1
f (ξ + m)e2πikξ dξ
0
f (ξ)e2πikξ dξ.
−∞
Z
e−2πikx
k=−∞
∞
f (ξ)e2πikξ dξ,
(7)
−∞
which is the Poisson summation formula.
Now we set f (x) = |x + iy|−2s = |z|−2s for x in (−∞, ∞). Then we see
∞
P
|m + z|−2s converges absolutely, uniformly in every interval
that the series
m=−∞
−N ≤ x ≤ N, for σ > 1/2. For this purpose, it clearly suffices to consider the
interval 0 ≤ x ≤ 1, since the series remains unchanged when x is replaces by
x + 1. For 0 ≤ x ≤ 1,
∞
∞
X
X
−2s
|m + z|−2σ
|m + z| ≤
m=−∞
m=−∞
∞
X
−2σ
−2σ
≤ |z|
+ |z − 1|
+
(m−2σ + m−2σ )
m=1
−2σ
< 2y
+2
∞
X
−2σ
m
m=1
!
1
.
σ>
2
Thus, by (7), for σ > 1/2,
∞
X
m=−∞
−2s
|m + z|
=
=
∞
X
k=−∞
∞
X
−2πikx
e
e−2πikx
k=−∞
= y1−2s
∞
X
k=−∞
Z
Z
∞
|ξ + iy|−2s e2πikξ dξ
−∞
∞
(ξ2 + y2 )−s e2πikξ dξ
−∞
e−2πikx
Z
∞
(1 + ξ2 )−s e2πikξ dξ,
(8)
−∞
whenever the series (8) converges. Actually, we shall prove that it converges
R∞
absolutely for σ > 1/2. For this purpose, let us consider for k > 0, −∞ (1 +
7
Kronecker’s Limit Formulas
7
ζ 2 )−s e2πikζ dζ as a line integral in the complex ζ-plane; let ζ = ξ + iη. The
function (1 + ζ 2 )−s has a logarithmic branch-point at ζ = i and ζ = −i. In
the complex ζ-plane cut along the η-axis from η = 1 to η = ∞ and from
η = −1 to η = −∞, let ABCDEFG denote the contour consisting of the straight
line GA(η = 0, |ξ| ≤ R), the arc AB(ζ = Reiθ π ≥ θ ≥ π/2), the straight line
BC(ξ = 0, 3/2 ≤ η ≤ R) on the left bank of the cut, the circle
CDE(ζ = i + (i/2)eiϕ , 0 ≤ ϕ ≤ 2π),
the straight line EF on the right bank of the cut (ξ = 0, 3/2 ≤ η ≤ R)
Fig. 1
andthe arc FG(ζ = Reiθ , π/2 ≥ θ ≥ 0). If s lies in a compact set K of the
s-plane with σ > 0, then on the arcs AB and FG,
|(1 + ζ 2 )−s e2πikyζ | ≤ |1 + ζ 2 |−σ e−2πky R sin θ
= 0(R−2σ e−2πky R sin θ ),
since −π ≤ arg(1 + ζ 2 ) ≤ π on the whole contour. The constants in the Oestimate depend only on K. Thus
!
Z π/2
Z
= 0 R−2σ
−2πky R sin θ
2 −s 2πikyζ
e
Rdθ
(1
+
ζ
)
e
dζ
0
AB
1−2σ
=0 R
Z
0
π/2
−4kyRθ
e
dθ
!
= 0(R−2σ ),
R
and hence AB (1 + ζ 2 )−s e2πikζ dζ tends to zero as R tends to infinity. Similarly
R
(1 + ζ 2 )−s e2πikζ dζ also tends to zero as R tends to infinity. If Γ+ denotes the
FG
8
Kronecker’s Limit Formulas
8
contour BCDEF in the limiting position as R tends to infinity, then for σ > 0,
we have, by Cauchy’s Theorem,
Z ∞
Z
(1 + ζ 2 )−s e2πikyζ dζ =
(1 + ζ 2 )−s e2πikyζ dζ.
(9)
Γ+
−∞
Let Γ− denote the limiting position of B′C ′ D′ E ′ F ′ (which is the reflection
of BCDEF with respect to ξ-axis), as R tends to infinity. We can show similarly
that when k < 0,
Z ∞
Z
(1 + ζ 2 )−s e2πikyζ dζ =
(1 + ζ 2 )−s e2πikyζ dζ.
(10)
Γ−
−∞
We shall now show that the right hand side of (??) defines an entire function
of s. In fact, if s = σ+it lies in a compact set K in the s-plane with σ > −d(d >
0), then, on Γ+ we have for a constant c1 depending only on K, |1 + ζ 2 |−σ ≤
c1 η2d . Thus for k > 0,
Z
Z
(1 + ζ 2 )−s e2πikyζ dζ ≤
|(1 + ζ 2 )−s |e−2πkyη |dζ|
Γ+
Z
Γ+
2 −σ π|t|−2πkyη
Z
|1 + ζ | e
|dζ| ≤ c1
η2d eπ|t|−2πkyη |dζ|
Γ+
Z
Z
1
1
≤ c2 e−πky
η2d e−2πky(η− 2 ) |dζ| ≤ c2 e−πky
η2d e−2πy(η− 2 ) |dζ|
≤
Γ+
Γ+
−πky
≤ c3 e
Γ+
,
(11)
R
the constants c2 and c3 depending only on K. Hence Γ+ (1 + ζ 2 )−s e2πkyζ dζ converges absolutely and uniformly in K and therefore defines an analytic function
of s in any domain contained in K. Since d is an arbitrary positive number, our
assertion above is proved. Similar to (11), we have also for k < 0,
Z
(1 + ζ 2 )−s e2πikyζ dζ ≤ c′ e−π|k|y ,
(12)
3
Γ−
R
and therefore Γ− (1 + ζ 2 )−s e2πikyζ dζ defines for k < 0, an entire function of s.
In view of (??), (10), (11) and (12), we see that in order to prove the absolute convergence
of the series (8), for σ > 1/2, we need to prove only that for
R∞
σ > 1/2, −∞ (1 + ζ 2 )−s dζ exists. But, in fact, for σ > 1/2,
Z
∞
−∞
2 −s
(1 + ζ ) dζ = 2
Z
∞
0
(1 + ζ 2 )−s dζ
9
Kronecker’s Limit Formulas
=
9
Z
∞
1
(1 + λ)−s λ− 2 dλ
(setting ζ 2 = λ)
0
Z
1
1
µ
µ− 2 (1 − µ) s−3/2 dµ setting λ =
1
−
µ
0
!
1
1
= B ,s−
,
2
2
=
!
where B(a, b) is Euler’s beta-function. Hence
Z
1
π 2 Γ(s − 12 )
(1 + ζ ) dζ =
Γ(s)
−∞
∞
2 −s
Thus, for σ > 1,
∞
X
m=−∞
|m + z|−2s =
1
Z ∞
∞
X
π 2 Γ(s − 21 ) 1−2s
′ −2πimx
y
+ y1−2s
e
(1 + ζ 2 )−s e2πimyζ dζ,
Γ(s)
−∞
m=−∞
where the accent on Σ indicates the omission of m = 0. Substituting this in (6),
we have for σ > 1,
ζQ (s) = 2y s ζ(2s)+

∞  21
1
X

 π Γ(s − 2 ) 1−2s 1−2s
s
+ 2y
y
+ n1−2s y1−2s ×

 Γ(s) n

n=1

Z ∞
∞

X

′ −2πimnx

2 −s 2πimnyζ
e
(1 + ζ ) e
dζ 
×


−∞
m=−∞
By (??) and (10), therefore, for σ > 1,
1
π 2 Γ(s − 12 )
ζ(2s − 1)+
ζQ (s) = 2y ζ(2s) + 2y
Γ(s)
Z ∞
∞
∞
X
X
e−2πimnx
(1 + ζ 2 )−s e2πimnyζ dζ;
+ 2y1−s
n1−2s
s
1−s
m=−∞
n=1
−∞
i.e.
1
∞
X
π 2 Γ(s − 21 )
ζQ (s) = 2y ζ(2s) + 2y
ζ(2s − 1) + 2y1−s
n1−2s ×
Γ(s)
n=1
∞
Z

X


×
(1 + ζ 2 )−s e2πimnyζ dζ+
e−2πimnx


+
Γ
s
1−s
m=1
10
Kronecker’s Limit Formulas
∞
X
2πimnx
e
m=1
10
Z
2 −s −2πimnyζ
(1 + ζ ) e
Γ−




dζ 
.


(13)
The double series in (13) converges absolutely, uniformly in every compact
subset of the s-plane, in view of the fact that by (11) and (12), it is majorised
by
 ∞

∞
∞
X
X
 X −πmny

1−σ
1−2σ 
′
−πmny
2y
n
e
+ c3
e
c3
 ,
n=1
m=1
m=1
whichclearly converges. Since we have seen earlier that the terms of the double
series define entire functions of s, it follows by the theorem of Weierstrass that
the double series in (13) defines an entire function of s.
Formula (13) gives us an analytic continuation of ζQ (s) for σ > 1/2. For,
ζ(2s) is regular for σ > 1/2 and from the fact that ζ(s) can be analytically
continued into the half-plane σ > 0 such that ζ(s)−1/(s−1) is regular for σ > 0,
we see on replacing s by 2s − 1 that ζ(2s − 1) can be continued analytically
in the half-plane σ > 1/2 such that ζ(2s − 1) − 1/2(s − 1) regular for σ >
1/2 : Γ(s − 12 )/Γ(s) is regular for σ > 1/2 and non-zero at s = 1. Hence from
(13), we see that ζQ (s) can be continued analytically in the half-plane σ > 1/2
and its only singularity in this half-plane is a simple pole at s = 1 with residue
1
1
equal to 2 · π 2 · π 2 · 1/2 = π.
ζQ (s) − π/(s − 1) has therefore a convergent power-series expansion a0 +
a1 (s − 1) + · · · at s = 1. We shall now determine the constant a0 . Clearly, from
(13),


1

 1−s π 2 Γ(s − 21 )
π
ζ(2s − 1) −
a0 = 2yζ(2) + lim 2y
 +
s→1
Γ(s)
s − 1
∞
Z
∞
X
1 X −2πimnx
+2
(1 + ζ 2 )−1 e2πimnyζ dζ+
 e
+
n
Γ
n=1
m=1

Z
∞
X

+
e2πimnx
(1 + ζ 2 )−1 e−2πimnyζ dζ 
m=1
Now from
ζ(s) =
we have
ζ(2s − 1) =
Γ−
1
+ C + a1 (s − 1) + · · ·
s−1
1
+ C + 2a1 (s − 1) + · · · .
2(s − 1)
11
Kronecker’s Limit Formulas
11
Also replacing s by s − 1/2 in the formula
!
√
1
Γ(s) = π · 21−2s Γ(2s).
Γ s+
2
wehave
Γ s−
!
√
1
Γ(s) = π22−2s Γ(2s − 1).
2
At s = 1, Γ(s) has the expansion
Γ(s) = 1 + a(s − 1) + · · ·
and hence
Γ2 (s) = 1 + 2a(s − 1) + · · ·
Further at s = 1,
Γ(2s − 1) = 1 + 2a(s − 1) + · · ·
Hence we have at s = 1, by Cauchy multiplication of power-series,
Γ(2s − 1)Γ−2 (s) = 1 + b(s − 1)2 + · · ·
and thus
1
2y1−s π 2
Γ(s − 21 )
Γ(2s − 1)
√
ζ(2s − 1)
ζ(2s − 1) = 2π(2 y)2−2s
Γ(s)
Γ2 (s)
= 2π(1 + b(s − 1)2 + · · · )×
√
× (1 − 2 log(2 y)(s − 1) + · · · )×
!
1
+C + ··· .
×
2(s − 1)
In other words,


 1−s 1 Γ(S − 21 )
π 
√
2

 = 2π(C − log(2 y))/
lim 2y π
ζ(2s − 1) −
s→1
Γ(s)
s−1
Now (see Fig. 1), for mn > 0, since (1 + ζ 2 )−1 has no branch-point at ζ = i or
ζ = −i,
Z
Z
(1 + ζ 2 )−1 e2πimnyζ dζ =
(1 + ζ 2 )−1 e2πimnyζ dζ
Γ+
CDE
!
e2πimnyζ
= 2πi Residue of
at
ζ
=
i
1 + ζ2
12
Kronecker’s Limit Formulas
12
= πe−2πmny .
Similarly, for mn > 0,
!
Z
e−2πimnyζ
2 −1 −2πimnyζ
at ζ = −i
(1 + ζ ) e
dζ = −2πi Residue of
1 + ζ2
Γ−
13
= πe−2πmny
Hence
a0 =
∞
∞
X
π2
1 X −2πimnx−2πmny
√
y + 2π(C − log(2 y)) + 2π
e
+
3
n m=1
n=1
+ 2π
∞
∞
X
1X
n=1
n m=1
e2πimnx−2πmny
These series converge absolutely and it is practical to sum over n first. Then
a0 =
∞
∞ X
X
1 −2πimnz
π2
√
y + 2π(C − log 2 y) + 2π
e
+
3
n
m=1 n=1
+ 2π
∞
∞ X
X
1
m=1 n=1
n
e2πimnz
∞
X
π2
√
= y + 2π(C − log(2 y)) − 2π
log(1 − e−2πimz )−
3
m=1
− 2π
∞
X
m=1
log(1 − e2πimz )
Y
√
= 2π(C − log(2 y)) − 2π(log e(πi/12)(z−z) + log
(1 − e2πimz )+
∞
m=1
+ log
∞
Y
m=1
(1 − e−2πimz )).
For complex z = x + iy with y > 0, Dedekind defined the η-function,
η(z) = eπiz/12
∞
Y
(1 − e2πimz ).
m=1
Inthe notation of Dedekind, then,
14
Kronecker’s Limit Formulas
13
√
a0 = 2π(C − log 2 y) − 2π log η(z)η(−z))
√
= 2π(C − log 2 − log( y|η(z)|2 )).
Thus we have proved
Theorem 1. Let z = x + iy, y > 0 and let Q(u, v) = y−1 (u + vz) · (u + vz).
Then the zeta function ζQ (s) associated with Q(u, v) and defined by ζQ(s) =
P′
(Q(m, n))−s , s = σ + it, σ > 1, can be continued analytically into a function
m,n
of s regular for σ > 1/2 except for a simple pole at s = 1 with residue π and at
s = 1, ζQ (s) has the expansion
ζQ (s) −
π
√
= 2π(C − log 2 − log( y|η(z)|2 )) + a1 (s − 1) + · · ·
s−1
This leads us to the interesting first limit formula of Kronecker, viz.
π √
lim ζQ (s) −
= 2π(C − log 2 − log y|η(z)|2 ).
s→1
s−1
We make a few remarks. It is remarkable that the residue of ζQ (s) at s = 1
does not involve a, b, c and this was utilized by Dirichlet in determining the
class-number of positive binary quadratic forms. Of course, we had supposed
that d √= ac − b2 = 1; in the general case, the residue of ζQ (s) at s = 1 would
be π/ d.
This limit formula has several applications; Kronecker himself gave one,
namely that of finding solutions of Pell’s (diophantine) equation x2 − dy2 = 4,
by means of elliptic functions. It has sereval other applications and it can
be generalized in many ways. In the next section, we shall show that as an
application of this formula, the transformation-theory of η(z) under the elliptic
modular group can be developed.
2 The Dedekind η-function
Let H denote the complex upper halt-plane, namely the set of z = x + iy with
y > 0. For z ∈ H, Dedekind defined the η-function
πiz/12
η(z) = e
∞
Y
(1 − e2πimz ).
m=1
15
This infinite product converges absolutely, uniformly in every compact subset of H. Thus, as a function of z, η(z) is regular in H. Since none of the factors
Kronecker’s Limit Formulas
14
of the convergent infinite product is zero in H, it follows that η(z) , 0, for z
in H. If g2 and g3 are the usual constants occurring in Weierstrass’ theory of
elliptic functions with the period-pair (1, z), then (2π)12 η24 (z) = g32 − 27g23 . The
function η(z) is a “modular form of dimension −1/2”.
Let α, β, γ, δ be rational integers such that αδ − βγ = 1. The transformation
z → z∗ = (αz + β)(γz + δ)−1 takes H onto itself; for, if z∗ = x∗ + iy∗ , then
!
1
1 αz + β αz + β
y∗ = (z∗ − z∗ ) =
−
2i
2i γz + δ γz + δ
1
= (z − z)|γz + δ|−2 = y|γz + δ|−2 > 0,
(14)
2i
and clearly z = (δz∗ −β)(−γz∗ +α)−1 . These transformations are called “modular
transformations” and they form a group called the “elliptic modular group”. It
is known that the elliptic modular group is generated by the simple modular
transformations, z → z + 1 and z → −1/z. In other words, any general modular
transformation can be obtained by iterating the transformations z → z ± 1 and
z → −1/z.
We shall give, in this section, two proofs of the transformation-formula for
the behaviour of η(z) under the modular transformation z → −1/z. The first
proof is a consequence of the Kronecker limit formula proved in § 1.
With z = x + iy ∈ H, we associate the positive-definite binary quadratic
form Q(u, v) = y−1 (u + vz)(u + vz). By the Kronecker limit formula for ζQ (s) =
∞
P
′
(Q(m, n))−s
m,n=−∞
lim ζQ (s) −
s→1
π √
= 2π(C − log 2 − log( y|η(z)|2 )).
s−1
(15)
Let z∗ = (αz + β)(γz + δ)−1 = x∗ + iy∗ be the image of z under a modular transformation. Then with z∗ , let us associate the positive-definite binary quadratic
form
Q∗ (u, v) = y∗−1 (u + vz∗ )(u + vz∗ ).
Again, by the Kronecker limit formula for
ζQ∗ (s) =
lim ζQ∗ (s) −
s→1
Now, by (??),
∞
X
′
16
(Q∗ (m, n))−s ,
m,n=−∞
p
π = 2π(C − log 2 − log y∗ |η(z∗ )|2 ).
s−1
y∗−1 |m + nz∗ |2 = y−1 |γz + δ|2 |m + n(αz + β)(γz + δ)−1 |2
(16)
Kronecker’s Limit Formulas
15
= y−1 |m(γz + δ) + n(αz + β)|2
= y−1 |(mδ + nβ) + (mγ + nα)z|2 .
Since α, β, γ, δ are integral and αδ − βγ = 1, when m, n run over all pairs
of rational integers except (0, 0), so do (mδ + nβ, mγ + nα). Thus, in view of
absolute convergence of the series, for σ > 1,
X′
ζQ∗ (s) =
(y∗−1 |m + nz∗ |2 )−s
m,n
=
=
X′
m,n
(y−1 |(mδ + nβ) + (mγ + nα)z|2 )−s
m,n
(y−1 |m + nz|2 )−s ;
X′
i.e. for σ > 1,
δQ (s) = ζQ∗ (s)
(ζQ (s) is what is called a “non-analytic modular function of z”).
Since ζQ (s) and ζQ∗ (s) can be analytically continued in the half-plane σ >
1/2, ζQ (s) = ζQ∗ (s) even for σ > 1/2. Hence, from (15) and (16),
p
√
log( y|η(z)|2 ) = log( y∗ |η(z∗ )|2 );
i.e.
1
1
|η(z)|y 4 = |η(z∗ )|y 4 .
By(??), then, we have
17
!
p
αz + β = |η(z)|| γz + δ|.
η
γz + δ
(17)
Let us consider the function
f (z) =
η((αz + β)(γz + δ)−1 )
p
γz + δη(z)
√
where the branch of γz + δ is chosen as follows; namely, if γ = 0, then
α
√ = δ = ±1 and assuming without loss of generality that α = δ =√1, we choose
γz + δ = 1. If γ , 0, we might suppose that γ > 0 and then γz + δ is that
branch
whose argument always lies between 0 and π for z ∈ H. Since η(z) and
√
γz + δ never vanish in H, (17) means that the funciton f (z) which is regular
in H is of obsolute value 1. By the maximum modulus principle, f (z) = ǫ, a
constant of absolute value 1. We have thus
Kronecker’s Limit Formulas
16
Proposition 3. The Dedekind η-function
η(z) = eπiz/12
∞
Y
m=1
(1 − e2πimz ), z ∈ H
satisfies, under a modular transformation z → (αz + β) · (γz + δ)−1 the transformation formula
p
η((αz + β)(γz + δ)−1 ) = ǫ γz + δη(z)
with ǫ = ǫ(α, β, γ, δ) and |ǫ| = 1.
We shall determine ǫ for the special modular transformations, z → z + 1
and z → −1/z.
From the very definition of η(z),
η(z + 1) = eπi/12 η(z)
(18)
and so here ǫ = eπi/12 .
√
η(−1/z) = ǫ zη(z), we have η(i) =
√ Also if we set z = i in the√formula
−πi/4
ǫ iη(i). Since η(i) , 0, ǫ = 1/ i = e
.
√
Thusη(−1/z) = e−πi/4 zη(z). We rewrite this as
! r
z
1
η(z),
(19)
η − =
z
i
√
z/i being that branch taking the value 1 at z = i.
To determine ǫ in the general case, we only observe that any modular
transformation is obtained by iteration of the transformations z → z ± 1 and
z → −1/z. Since every time we apply these transformations we get, in view of
(18) and (19), a factor which is e±πi/12 or e−πi/4 , we see that ǫ is a 24th root of
unity; ǫ can be determined this way, by a process of “reduction”.
The question arises as to whether there exists an explicit expression for ǫ,
in terms of α, β, γ and δ. This problem was considered and solved by Dedekind
who for this purpose, had to investigate the behaviour of η(z) as z approaches
the ‘rational points’ on the real line. This problem has also been considered
recently by Rademacher in connection with the so-called ‘Dedekind sums’.
We now give a very simple alternative proof of formula (19).
18
Kronecker’s Limit Formulas
17
Fig. 2
Letus consider the integral
Z
1
πζ dζ
cot πζ cot
8 Ct
z ζ
19
where Ct is the contour of the parallelogram in the ζ-plane with vertices at
ζ = t, tz, −t and −tz (t positive and not integral).
The integrand is a meromorphic function of ζ. It has simple poles at ζ = ±k
πk
1
1
cot
and
cot πkz respecand ζ = ±kz(k = 1, 2, 3, . . .) with residues
πk
z
πk
tively, as one can readily notice from the expansion
cot πu =
1
πu
−
+ ···
πu
3
valid for 0 < |u| < 1. Also, the integrand has a pole of the third order at ζ = 0,
with residue − 31 (z + 1/z).
Let t = h + 1/2, h being a positive integer. Then there are no poles of
the integrand on Ct and inside Ct there are poles at ζ = 0, ±k and ±kz(k =
1, 2, . . . , h). By Cauchy’s theorem on residues,
 h
!
Z
h


1 
1
πk X′ 1
1
ζ dζ 2πi 

X′ 1
z+ 
,
=
cot
+
cot πkz −
cot πζ cot π



8 C 1
z ζ
8 
πk
z
πk
3
z 
h+ 2
k=−h
k=−h
the accent on Σ indicating the omission of k = 0 from the summation. In other
Kronecker’s Limit Formulas
18
words,
1
8
Z
Ch+ 1
2
 h
!
!


1
i
πk 
ζ dζ

X 1
z+
= 
cot πkz + cot
.
cot πζ cot π




z 12
z
2 k=1 k
z 
(20)
We wish to consider the limit of the relation (20) as h tends to infinity
through the sequence of natural numbers.
For this purpose, let us first observe that if ζ = ξ + iη, then
cot πζ = i
eπiζ + e−πiζ
eπiζ − e−πiζ
tendsto −i if η tends to ∞, and tends to +i, if η tends to −∞. Similarly cot π(ζ/z)
tends to −i, when the imaginary part of ζ/z tends to ∞ and to i, when the latter
tends to −∞. If O is the origin and ζ tends to infinity along a ray OP with
P on one of the open segments AB, BC, CD, DA of Ch+ 12 (see figure), then
cot πζ · π(ζ/z) tends to the values +1, −1, +1, −1 respectively. Moreover, this
process of tending to the respective values is uniform, when the ray OP lies
between two fixed rays from O, which lie in one of the sectors AOB, BOC,
COD or DOA and neither of which coincides with the lines η = 0 or ζ = λz
(λ real). This means that the sequence of functions {cot π(h + 1/2)ξ cot π(h +
1/2)(ζ/z)} (h = 1, 2, . . .) tends to the limits +1 or −1 uniformly on any proper
closed subsegment of a side of C1 .
Moreover, this sequence of functins is uniformly bounded on C1 . This
can be seen as follows. Let K j denote the discs |ζ − j| < 1/4, j = 0, ±1,
±2, . . .. In view of the periodicity of cot πζ, let us confine ourselves to the
strip, 0 ≤ ξ ≤ 2. We then readily see that in the complement of the setunion of the discs K0 , K1 and K2 with respect to this vertical strip, the function
cot πζ is bounded; for cot πζ tends to πi as η tends to ±∞ and is bounded away
from its poles at ζ = 0, 1 and 2. Indeed therefore, cot πζ is bounded in the
complement of the set union of the discs K j , j = 0, ±1, . . . with respect to the
entire plane. Since the contours Ch+ 21 , for h = 1, 2, . . . lie in the complement,
we have | cot πζ| ≤ α for ζ ∈ Ch+ 21 and α independent of h and ζ. By a similar
argument for cot(πζ/z) concerning its poles at ±kz, we see that for ζ ∈ Ch+ 12 ,
h = 1, 2, . . . | cot(πζ/z)| ≤ β, β independent of h and ζ. In other words, the
sequence of functions {cot π(h + 12 )ζ cot π(h + 21 )(ζ/z)} is uniformly bounded on
c1 . Now
!
!
Z
Z
1 ζ dζ
1
πζ dζ
ζ cot π h +
=
,
(21)
cot π h +
cot πζ cot
z ζ
2
2 z ζ
C1
Ch+ 1
2
20
Kronecker’s Limit Formulas
19
and in view of the foregoing considerations, we can interchange the passage to
the limit (as h tends to infinity) and the integration, on the right side of (21).
Thus, letting h tend to infinity, we obtain from (20),
!
Z z Z −1 Z −z Z 1 !
πi
1
1
dζ
z+
+
−
+
−
12
z
8 1
ζ
z
−1
−z
∞
!


X
1
πk 
i


cot πkz + cot
.
(22)
= 




2 k=1 k
z 
Now
1
8
Z
1
z
−
Z
−1
z
+
Z
−z
−1
−
Z 1!
−z
dζ 1
=
ζ
4
Z
z
+
1
Z z!
−1
dζ
.
ζ
For z in the ζ-plane cut along the negative ξ-axis, let log z denote the branch
that is real on the positive ξ-axis. Then we see
Z z
Z z
dζ
dζ
= log z and
= log z − πi.
1 ζ
−1 ζ
Thus from (22),
∞
!
!

1
1
πk 
πi
πi i X 1

z+
+ log z −
cot πkz + cot
.
= 


12
z
2
2
2 k=1 k
z 
(23)
We insert in (23), the expansions
and


!
∞
X


e2πikz
2πimkz

cot πkz = −i 1 + 2
= −i 1 + 2
e
2πikz
1−e
m=1


!
∞
X


e−2πik/z
πk
−2imπk/z
 ;
= i 1 + 2
e
=i 1+2
cot
−2πik/z
z
1−e
m=1
then we see that the resulting double series is absolutely convergent. We are
therefore justified in summing over k first; we see as a result that the series (23)
goes over into
!
1
!
η −
∞
X
z
1
πi
(1 − e2πimz )
z
+
.
+
=
log
−
log
η(z)
12
z
(1 − e−2πim/z )
m=1
21
Kronecker’s Limit Formulas
Finally
20
η − 1z
r
z
1
πi log
=
,
log z −
= log
η(z)
2
2
i
√
where
z/i is that branch which takes the value 1 at z = i i.e. η(−1/z) =
√
(z/i)η(z).
We wish to remark that it was only for the sake of simplicity that we took
a parallelogram C1 and considered ‘dilatations’ Ch+ 21 of C1 . It is clear that
instead of the parallelogram, we could have taken any other closed contour
passing through 1, z, −1, −z and through no other poles of the integrand and
worked with corresponding ‘dilatations’ of the same.
22
3 The second limit formula of Kronecker
We Shall consider now some problems more general than the ones concerning
the analytic continuation of the function ζQ (s) in the half-plane σ > 1/2 and
the Kronecker limit formula for ζQ (s).
In the first place, we ask whether it is possible to continue ζQ (s) analytically
in a larger half-plane; this question shall be postponed for the present. We shall
show later (§ 5) that ζQ (s) has an analytic continuation in the entire plane which
is regular except for the simple pole at s = 1 and satisfies a functional equation
similar to that of the Riemann ζ-function.
∞
P
′
(Q(m, n))−s , one might also consider
Instead of ζQ (s) =
m,n=−∞
∞
X
(Q(m + µ, n + ν))−s
m,n=−∞
where µ and ν are non-zero real numbers which are not both integral and where
m and n run independently from −∞ to +∞. Or, instead of the positive-definite
binary quadratic form Q(u, v), we might take a positive-definite quadratic form
in more than two variables. Let, in fact, S = (si j ), 1 ≤ i, j ≤ p be the matrix of
P
a positive-definite quadratic form
si j xi x j in p variables. Let us consider
1≤i, j≤p
ζS (S ) =
∞
X
x1 ,...,x p =−∞


X

 S i j xi x j 
i, j
x1 , . . . , x p running over all p-tuples of integers, except (0, . . . , 0). The functionζS (S ), known as the Epstein zeta-function, is regular for σ?p/2. It was
23
Kronecker’s Limit Formulas
21
shown by Epstein that ζS (S ) can be continued analytically in the entire s-plane,
into a function regular except for a simple pole at S = p/2 and satisfying the
functional equation
p
p
1
π−s Γ(S )ζS (S ) = |S |− 2 π−(p/2−s) Γ
− s ζS −1
−s
2
2
Epstein also obtained for ζS (s), an analogue of the first limit formula of Kronecker but this naturally involves more complicated functions than η(z).
We shall now consider a generalization of ζQ (s) which shall lead us to the
second limit formula of Kronecker. namely, let u and v be independent real
parameters. Let, as before, Q(m, n) = am2 + 2bmn + cn2 be a positive-definite
binary quadratic form and let ac − b2 = 1, without loss of generality. We then
define, for σ > 1,
X′
e2πi(mu+nv) (Q(m, n))−s
(24)
ζQ (s, u, v) =
m,n
m, n running over all ordered pairs of integers except (0, 0). The series converges absolutely for σ > 1 and converges uniformly in every half-plane
σ ≥ 1 + ǫ(ǫ > 0). Thus ζQ (s, u, v) is a regular function of s for σ > 1. If
u and v are both integers, then ζQ (s, u, v) = ζQ (s) which has been already considered. We shall, suppose, in the following, that
u and v are not both integers.
(∗)
Furthermore, on account of the periodicity of ζQ (s, u, v) in u and v, we can
clearly suppose that 0 ≤ u, v < 1. The condition (∗) then means that at least
one of the two conditions 0 < u < 1, 0 < v < 1 is satisfied. We might, without
loss of generality, suppose that 0 < u < 1. For, otherwise, if u = 0, then
necessarily 0 < v < 1. In this case, let us take the positive-definite binary
quadratic form Q1 (m, n) = cm2 + 2bmn + an2 ; we see then that, for σ > 1,
X′
e2πi(mv+nu) (Q1 (m, n))−s .
ζQ (s, u, v) =
m,n
And here, since 0 < v < 1, v shall play the role of u in (24).
Weshall prove now that ζQ (s, u, v) can be continued analytically into a function regular in the half-plane σ > 1/2 and then determine its value at s = 1;
this will lead us to the second limit formula of Kronecker.
For σ > 1, by our earlier notation,
ζQ (s, u, v) = y s
∞
X
m=−∞
e2πimu |m|−2s + y s
∞
X
n=−∞
e2πinv
∞
X
m=−∞
e2πimu |m + nz|−2s . (25)
24
Kronecker’s Limit Formulas
22
The first series in (25) converges absolutely for σ > 1/2 and uniformly in
every compact subset of this half-plane. Hence it defines a regular function of
s for σ > 1/2. The double series in (25) again defines a regular function of
s in the half-plane σ > 1. To obtain its analytic continuation in a larger halfplane, we shall carry out the summation of the double series in the following
particular way. Namely, we consider the finite partial sums
X
X′
e2πi(mu+nv) |m + nz|−2s .
n1 ≤n≤n2 m1 ≤m≤m2
We shall show that when m1 and n1 tend to −∞ independently and m2 and n2
tend to ∞ independently, then these partial sums which are entire functions of
s converge uniformly in every compact subset of the half-plane σ > 1/2. The
proof is based on the method of partial summation in Abel’s Theorem.
e2πiku
. (Recall that 0 < u < 1).
Let us define for any integer k, ck = 2πiu
e
−1
2πimu
Then e
= cm+1 − cm . Moreover,
|cm | ≤ |1 − e2πiu |−1 = α,
where α is independent of m. Now
X
X′
e2πi(mu+nv) |m + nz|−2s
n1 ≤n≤n2 m1 ≤m≤m2
m2
n2
X
X
2πinv
=
e
n=n1
m=m1
cm+1 − cm
|m + nz|2s
 m
2
 X
2πinv 

=
e
cm (|m − 1 + nz|−2s − |m + nz|−2s )+

n=n1
m=m1 +1


−2s
−2s 
+ cm2 +1 |m2 + nz| − cm1 |m1 + nz|  .
n2
X
Let us also observe that
cm2 +1 |m2 + nz|−2s ≤ α|n|−2σ y−2σ
and
Further
cm1 |m1 + nz|−2s ≤ α|n|−2σ y−2σ .
||m − 1 + nz|−2s − |m + nz|−2s |
(26)
25
Kronecker’s Limit Formulas
Z
−2s
23
((µ + nx)2 + n2 y2 )−s−1 (µ + nx)dµ
m−1
Z m
1
≤ 2|s|
((µ + nx)2 + n2 y2 )−(σ+ 2 ) du.
m
m−1
Thus (26) is majorised by
Z m
n2
n2
m2
X
X
′ X
′
1
((µ + nx)2 + n2 y2 )−(σ+ 2 ) dµ + 2αy−2σ
2|s|α
|n|−2σ .
n=n1 m=m1 +1
m−1
(27)
n=n1
In (27), we sum m from −∞ to +∞ instead of summing from m1 + 1 to m2 and
then we see that (26) has the majorant
n2 Z ∞
n2
X
X
′
1
|n|−2σ .
2α|s|
((µ + nx)2 + n2 y2 )−(σ+ 2 ) dµ + 2αy−2σ
n=n1
Now
Z
∞
−∞
n=n1
((µ + nx)2 + n2 y2 )
−∞
−(σ+ 21 )
Z
26
∞
−(σ+ 21 )
dµ
(µ2 + n2 y2 )
Z ∞
1
= |n|−2σ y−2σ
(1 + µ2 )−(σ+ 2 ) dµ.
dµ =
−∞
(28)
−∞
Since the integral in (28) converges for σ > 1/2, we see that (26) is majorised
n2
P
|n|−2σ , β depending only on the compact set in which s lies in the
by β
n=n1
half-plane σ > 1/2 and on α and y.
If, in (26), we now carry out the passage of m1 and n1 to −∞ and of m2
and n2 to +∞ independently, then we see that the double series in (25) summed
∞
P
in this particular manner, is majorised by the series 2β n−2σ . Hence it conn=1
verges uniformly when s lies in a compact set in the half-plane σ > 1/2. Since
each partial sum (26) is an entire function of s, we see, by Weierstrass’ Theorem that the double series in (25) converges uniformly to a function which is
regular for σ > 1/2 and which provides the necessary analytic continuation in
this half-plane.
Thus under the assumption that u and v are not both integers, ζQ (s, u, v) has
an analytic continuation which is regular for σ > 1/2. We shall now determine
its value at s = 1. We wish to make it explicit that hereafter we shall make only
the assumption (∗) and not the specific assumption 0 < u < 1.
From the convergence proved above, it follows that
ζQ (1, u, v) =
∞
z − z X′ e2πimu
+
2i m=−∞ |m|2
Kronecker’s Limit Formulas
+
24
∞
∞
z − z X′ 2πinv X
e2πimu
e
2i n=−∞
(m + nz)(m + nz)
m=−∞
(29)
where the accents on Σ have the usual meaning.
∞ e2πimu
∞ e2πimu
P
P
, we observe that since the series
In order to sum
2
m
m=−∞
m=−∞ |m|
converges uniformly to 2π(1/2 − u) in any closed interval. 0 < p ≤ u ≤ q < 1,
we have


Z u  X
Z u  X
∞
∞

′ e2πimu 
e2πimu 


 du

 du = lim

ǫ→0 ǫ
m
m 
0
m=−∞
m=−∞
∞ Z u 2πimu
X
′
e
= lim
du,
ǫ→0
m
ǫ
m=−∞
i.e.

 ∞
∞
 X′ e2πimu X
′ e2πimu 
1


−
2πi(u − u ) =
lim 
,
2 
2πi ǫ→0 m=−∞ m2
m
m=−∞
2
i.e.
2π2 (u2 − u) =
since
∞
P
′
m=−∞
∞
X
′
m=−∞
−
e2πimu π2
− ,
3
m2
2πimu
e
m2
is continuous in u. In other words,
!
∞
X
′ e2πimu
1
2
2
=
2π
u
−
u
+
.
6
m2
m=−∞
We shall sum the double series in (29) more generally with z replaced by r,
where −r ∈ H. Leter, we shall set r = z. We remark that with slight changes,
P
our arguments concerning the series ′ e2πi(mu+nv) |m + nz|−2s above will also
P′ 2πi(mu+nv)
e
((m + nz) · (m + nr))−s . Now for
go through for the double series
summing the series
∞
∞
z − r X′ 2πinv X
e2πimu
e
2i n=−∞
(m + nz)(m + nr)
m=−∞
=
!
∞
∞
1
1
1 X′ e2πinv X 2πimu
e
−
.
2i n=−∞ n m=−∞
m + nr m + nz
(30)
we could apply the Poisson summation formula with respect to m to the inner
sum on the left side of (30), but we shall adopt a different method here.
27
Kronecker’s Limit Formulas
25
Letus define, for complex z = x + iy,
f (z) =
28
!
∞
∞
1 X X e2πimu e−2πimu
+
+
+
z m=1 m=1 z + m
z−m
and consider f (z) in the strip −1/2 ≤ x ≤ 1/2. It is not hard to see that
whenever z lies in a compact set not containing z = 0 in this strip, the series
converges uniformly and hence f (z) is regular in this strip except at z = 0,
where it has a simple pole with residue 1. Moreover, one can show that f (z) −
1/z is bounded in this strip. Further
f (z + 1) = e−2πiu f (z).
Consider now the function
g(z) = 2πi
e−2πiuz
;
1 − e−2πiz
g(z) is again regular in this strip except at z = 0 where it has a simple pole with
residue 1. Moreover g(z) = 1/z is bounded in this strip; as a matter of fact, if
0 < u < 1, g(z) tends to 0 if y tends to +∞ or −∞ and if u = 0, then g(z) tends
to 0 or 2πi according as y tends to +∞ or −∞. Further g(z + 1) = e−2πiu g(z).
As a consequence, the function f (z) − g(z) is regular and bounded in this
strip and since f (z + 1) − g(z + 1) = e−2πiu ( f (z) − g(z)), it is regular in the whole
z-plane and bounded, too. By Liouville’s theorem, f (z) − g(z) = c, a constant.
It can be seen that, if u , 0, c = 0 and if u = 0, c = πi, by using the series
expansion for the function π cot πz. In any case,
f (nr) − f (nz) = g(nr) − g(nz).
(31)
In view of the convergence-process of the series (29) described above, we
can rewrite the double series (30) as

!
∞
∞



1 X′ e2πinv 
1 X 2πimu
1
1
1
1

1
−
+
+
−
−
e





2i n=−∞ n
nr nz m=1
m + nr −m + nr m + nz −m + hz 
!
∞
∞
X
′ e2πinv
e−2πiunr
e−2πiunz
1 X′ e2πinv
( f (nr) − f (nz)) = π
−
,
=
2i n=−∞ n
n
1 − e−2πinr 1 − e−2πinz
n=−∞
(32)
by (31). Now, we insert in (32), the expansions
 P
∞


− m=1 e2πimnz ,
−2πinz −1
(1 − e
) =
P∞ −2πmnz

 m=0 e
,
29
(n > 0)
(n < 0)
Kronecker’s Limit Formulas
26
−2πinr −1
(1 − e
)
P
∞


 m=0 e−2πimnr ,
=
P

2πimnr
− ∞
,
m=1 e
(n > 0)
(n < 0)
valid for z, −r ∈ H and then the series (32) goes over into

∞
∞
X
X

1
 2πin(v−ur)
−2πin(v−uz)
π
e
+
e
+
(e−2πin(v−uz−mz) +


n
n=1
m=1




+ e2πin(v−ur−mr) + e2πin(v−uz+m z) + e−2πin(v−ur+mr) )


′
(33)
If 0 < u < 1, then this series converges absolutely, since z, −r ∈ H . Carrying
out the summation over n first, we see that this double series is equal to
∞


Y
− π log 
(1 − e−2πi(v−uz−mz) )(1 − e2πi(v−ur−mr) )×


m=0

∞

Y

2πi(v−uz+mz)
−2πi(v−ur+mr) 
×
(1 − e
)(1 − e
)
.
(34)


m=1
If u = 0, then necessarily 0 < v < 1 and in this case,
π
∞
X
1
n=1
n
(e2πinv + e−2πinv ) = −π log(1 − e2πiv )(1 − e−2πiv ).
Therest of the series (33) is absolutely convergent and summing over n first,
we see that the value of the series (33) is given by the expression (34).
Let us now set v − uz = w, v − ur = q. Then replacing z by r in (29), we
have finally, for 0 ≤ u, v < 1 and u and v not simultaneously zero,
∞
e2πi(mu+nv)
z−r X
2i m,n=−∞ (m + nz)(m + nr)
∞
!


1
Y
2
2
= −π i(z − r) u − u +
− π log 
(1 − e−2πi(w−mz) )×


6
m=0
2πi(q−mr)
×(1 − e
)×
∞
Y
m=1
2πi(w+mz)
(1 − e
−2πi(q+mr)
)(1 − e




)


(35)
For z ∈ H and arbitrary complex w, the elliptic theta-function ϑ1 (w, z) is
defined by the infinite series,
ϑ1 (w, z) =
∞
X
n=−∞
1 2
1
1
eπi(n+ 2 ) z+2πi(n+ 2 )(w− 2 ) .
30
Kronecker’s Limit Formulas
27
It is clear that this series converges absolutely, uniformly when z lies in a compact set in H and w in a compact set of the w-plane. Hence ϑ1 (w, z) is an
analytic function of z and w for z ∈ H and complex w. It is proved in the
theory of the elliptic theta-functions that ϑ1 (w, z) can also be expressed as an
absolutely convergent infinite product, namely,
ϑ1 (w, z) = −ieπi(z/4) (eπiw − e−πiw )×
∞
Y
×
(1 − e2πi(w+mz) )(1 − e−2πi(w−mz) )(1 − e2πimz ).
(36)
m=1
We shall identify these two forms of ϑ1 (w, z) later on. For the present, we shall
consider ϑ1 (w, z) as defined by the infinite product.
Now a simple rearrangement of the expression (35) gives
∞
e2πi(mu+nv)
z − r X′
2i m,n=−∞ (m + nz)(m + nr)
!
!
1
1
2
2
2
= −π i(z − r) u − u +
+ π i (w − q) + (z − r) −
6
6
ϑ1 (w, q)ϑ1 (q, −r)
− π log
η(z)η(−r)
ϑ1 (w, q)ϑ1 (q, −r)
(w − q)2
− π log
= −π2 i
z−r
η(z)η(−r)
2πi(mu+nv)
X
′
z−r
ϑ1 (w, z)ϑ1 (q, −r) πi((w−q)2 /(z−r))
e
= log
e
.
−2πi m,n (m + nz)(m + nr)
η(z)η(−r)
(37)
Setting r = z again, we have q = w and then
ζQ (1, u, v) = −π2 i
31
ϑ1 (w, z)ϑ1 (w, −z)
(w − w)2
− π log
z−z
|η(z)|2
One can see easily from (36) that ϑ1 (w, z) = ϑ1 (w, −z). Hence
ϑ1 (w, z) ,
ζQ (1, u, v) = −π2 iu2 (z − r) − 2π log η(z) (38)
for 0 ≤ u, v < 1 and u and v not both zero.
We contend that formula (38) is valid for all u and v not both integral. Since
ζQ (1, u + 1, v) = ζQ (1, u, v + 1) = ζQ (1, u, v) and w goes over into w + 1 and
Kronecker’s Limit Formulas
28
w − z respectively under the transformations v → v + 1 and u → u + 1, it would
suffice for this purpose to prove that
ϑ1 (w + 1, z) ζQ (1, u, v) = −π2 iu2 (z − r) − 2π log η(z)
and
ϑ1 (w − z, z) .
ζQ (1, u, v) = −π2 i(u + 1)2 (z − r) − 2π log η(z)
The first assertion is an immediate consequence of the fact that ϑ1 (w + 1, z) =
−ϑ1 (w, z). To prove the second, we observe that from (36), we have
ϑ1 (w − z, z) (eπi(w−z) − e−πi(w−z) )(1 − e2πiw )
=
ϑ1 (w, z)
(eπiw − e−πiw )(1 − e−2πi(w−z) )
+2πiw−πiz
=e
and hence
32
!
ϑ1 (w − z, z) ϑ (w, z) − log 1
= +π2 i{(2(w − w) − (z − z)}
2π log η(z)
η(z) = −π2 i(z − z)((u + 1)2 − u2 ).
Thus the second assertion is proved and we have
Theorem 2. If u and v are real and not both integral, then the Epstein zetafunction ζQ (s, u, v) defined by (24) for σ > 1, can be continued analytically
into a function regular for σ > 1/2 and its value at s = 1 is given by (38).
We are finally led to the following formula: viz. for all u and v not both
integral,
2
z − z X′ e2πi(mu+nv)
ϑ1 (w, z) .
2 (w − w)
=
−π
i
−
2π
log
η(z) 2i m,n |m + nz|2
z−z
This is the second limit formula of Kronecker. We may rewrite it as
2
′
z − z X e2πi(mu+nv)
ϑ1 (v − uz, z) eπizu2 .
=
log
−2πi
η(z)
|m + nz|2
(39)
If u and v are both integral, then v − uz is a zero of ϑ1 (w, z) and then both
sides are infinite.
Kronecker’s Limit Formulas
29
4 The elliptic theta-function ϑ1 (w, z)
For z = x + iy ∈ H and arbitrary complex w, the elliptic theta-function ϑ1 (w, z)
was defined in § 3, by the infinite product (36). ϑ1 (w, z) is a regular function
of z and w, for z in H and arbitrary complex w. We shall now develop the
transformation-theory of ϑ1 (w, z) under modular transformations, as a consequence of the second limit formula of Kronecker.
Letu and v be arbitrary real numbers which are not simultaneously integral.
Let z → z∗ = (αz + β)(γz + δ)−1 = x∗ + iy∗ be a modular transformation. Then
u∗ = δu + γv and v∗ = βu + αv are again real numbers, which are not both
integral. For σ > 1, we have
y∗s
33
∗
∗
∞
X′
X
e2πi((mδ+nβ)u+(mγ+nα)v)
e2πi(mu +nv )
s
=
y
|m + nz∗ |2s
|(mδ + nβ) + (mγ + nα)z|2s
m,n
m,n=−∞
X′ e2πi(mu+nv)
= ys
.
|m + nz|2s
m,n
From § 3, we know that both sides, as functions of s, have analytic continuations regula in the half-plane σ > 1/2 and hence the equality of the two sides
is valid even for σ > 1/2. In particular, for s = 1,
y∗
X′ e2πi(mu∗ +nv∗ )
X′ e2πi(mu+nv)
=y
.
∗
2
|m + nz |
|m + nz|2
m,n
m,n
(40)
By Kronecker’s second limit formula,
2
∗ ∗ (w∗ − w∗ )2
ϑ1 (w , z ) = −π∗ i (w − w) − 2π log ϑ1 (w, z) −
2π
log
−π2 i ∗
η(z∗ ) η(z) z−z
z − z∗
where w = v − uz and w∗ = v∗ − u∗ z∗ = w/(γz + δ). Therefore,
ϑ1 (w, z) πi (ww)2 πi (w∗ − w∗ )2
ϑ1 (w∗ , z∗ ) =
− log −
log η(z∗ ) η(z) 2 z − z
2 z∗ − z∗
!
w2
w2
πiγ
2
= log e(πiγw )/(γz+δ) −
=
2 γz + δ γz + δ
Thus
ϑ1
αz+β
w
γz+δ , γz+δ
αz+β η γz+δ
=
ϑ1 (w, z) (πiγw2 )/(γz+δ)
e
· ω,
η(z)
(41)
where |ω| = 1 and ω might depend on every one of w, z, α, β, γ and δ. Butfrom
34
Kronecker’s Limit Formulas
30
(41), we observe that ω is a regular function of z in H and w, except possibly
when w is of the form m + nz with integral m and n. But since |ω| = 1, we see,
by applying the maximum modulus principle to ω as a function of w and of z,
that ω is independent of w and z. Hence ω = ω(α, β, γ, δ).
Now from the fact that
ϑ1 (w, z) = 2πη3 (z)w + · · ·
we note that ϑ′1 (0, z), the value of the derivative of ϑ1 (w, z) with respect to w at
w = 0, is equal to 2πη3 (z). Differentiating the relation (41) with respect to w at
w = 0, we get
αz+β ′
ϑ′ (0, z)
1 ϑ1 0, γz+δ
αz+β = 1
ω,
γz + δ η
η(z)
γz+δ
i.e.
(γz + δ)−1 η2
!
αz + β
= ωη2 (z).
γz + δ
We have thus rediscovered the formula
!
p
αz + β
η
= ǫ γz + δη(z)
γz + δ
with ǫ being a 24th root of unity depending only on α, β, γ and δ and ω = ǫ 2 .
Rewriting (41), we obtain
Proposition 4. The elliptic theta-function has the transformation formula
!
p
αz + β
w
2
,
= ǫ 3 γz + δe(πiγw )/(γz+δ) ϑ1 (w, z)
ϑ1
γz + δ γz + δ
under a modular substitution z → (αz + β)(γz + δ)−1 , ǫ 3 being an 8th root of
unity depending only on α, β, γ and δ.
From the transformation-theory of η(z) we know that corresponding to the
modular transformations z → z+1 and z → −1/z, ǫ 3 has respectively the values
eπi/4 and e−3πi/4 . Hence


ϑ1 (w, z + 1) = eπi/4 ϑ1 (w, z)



r

!
(42)

w 1
z

πiw2 /z 1

,− = e
ϑ1 (w, z).
ϑ1

z z
i
i
Once again using the fact that the modular transformations z → z + 1 and
35
Kronecker’s Limit Formulas
31
z → −1/z generate the elliptic modular group, we can determine ǫ 3 , with the
help of (42), by a process of ‘reduction’. Hermite found an explicit expression
for ǫ 3 , involving the well-known Gaussian sums.
We now wish to make another remark which is of function-theoretic significance and which is again, a consequence of Kronecker’s second limit formula.
Let us define for z ∈ H and real u and v (not both integral), the function
!
ϑ1 (v − uz, z) πizu2
e
f (z, u, v) = log
η(z)
choosing a fixed branch of the logarithm in H. Then from (40) and (39),
ϑ1 (v∗ − u∗ z∗ , z∗ ) πiz∗ u∗2 ϑ1 (v − uz, z) eπizu2 .
=
log
log e
η(z∗ )
η(z)
Hence
!
αz + β
f
, δu + γv, βu + αv = f (z, u, v) + iλ,
γz + δ
where λ is real and might depend on z, u, v, α, β, γ and δ. But since λ is a
regular function of z, whose imaginary part is zero in H, λ is independent of z.
Thus
!
αz + β
f
(43)
, δu + γv, βu + αv = f (z, u, v) + iλ(u, v, α, β, γ, δ).
γz + δ
Let now q > 1, be a fixed integer and let u and v be proper rational fractions
with reduced denominator q i.e. u = a/q, v = a/q, (a, b, q) = 1 and 0 ≤ u,
v < 1. Since αδ − βγ = 1, we have again.
(δa + γb, βa + αb, q) = 1.
ϑ1 (v − uz, z) πizu2 is invariant under the
Nowfrom (39), we see that log e
η(z)
transformations u → u + 1 and v → v + 1. Hence f (z, u, v) picks up a purely
imaginary additive constant under these transformations. If now, (δa + γb)/q ≡
(a∗ /q)( mod 1) and (βa + αb)/q ≡ (b∗ /q)( mod 1) where a∗ /q, b∗ /q are
proper rational fractions with reduced denominator q, then
!
!
αz + β δa + γb βa + γb
αz + β a∗ b∗
f
= f
+ iλ′
,
,
, ,
γz + δ
q
q
γz + δ q q
where λ′ is real and depends on a, b, α, β, γ and δ. Writing f (z, a/q, b/q) as
fa,b (z), we get from (43),
!
αz + β
= fa,b (z) + iλ∗ (a, b, α, β, γ, δ)
(44)
fa∗ ,b∗
γz + δ
36
Kronecker’s Limit Formulas
32
λ∗ (a, b, α, β, γ, δ) being a real constant depending on a, b, α, β, γ and δ.
The number of reduced fractions a/q, b/q with reduced denominator q is
Q
given by v(q) = q2 (1−1/p2 )p running through all prime divisors of q. Correp|q
sponding to each pair a/q, b/q, we have a function fa,b (z) and the relation (44)
asserts that when we apply a modular transformation on z, these functions are
permuted among themselves, except for a purely imaginary additive constant.
The modular transformations z to(αz + β)(γz + δ)−1 for which αγ βδ ≡ 10 01 (
mod q), form a subgroup M (q) of the elliptic modular group called the principal congruence subgroup of level (stufe) q.
The group M (q) has index qv(q) in the elliptic modular group and acts discontinuously on H. We can construct in the usual way, a fundamental domain
for M (q) in H. This can be made into a compact Riemann surface by identifying the points on the boundary which are ‘equivalent’ under M (q) and ‘adding
the cusps’. Complex-valued functions f (z) which are meromorphic in H and
invariant under the modular transformations in M (q) and have at most a pole
in the ‘local uniformizer’ at the ‘cusps’ of the fundamental domain are called
modular functions of level q. They form an algebraic function field of one
variable (over the field of complex numbers) which coincides with the field of
meromorphic functions on the Riemann surface.
Inthe case when the transformation z → (αz + β)(γz + δ)−1 is in M (q), then
referring to (44), we have a∗ = a, b∗ = b and
!
αz + β
= fa,b (z) + iλ∗ (a, b, α, β, γ, δ).
(45)
fa,b
γz + δ
The function fa,b (z) is regular everywhere in H. To examine its singularities
(in the local uniformizers) at the ‘cusps’ on the Riemann surface, it suffices
in virtue of (44) to consider the singularity of fa,b (z) at the ‘cusp at infinity’.
It can be seen that at the ‘cusp at infinity’, fa,b (z) has the singularity given by
2 −2
−1
log eπiz(a q −aq +1/6) . In view of (45), then fa,b (z) is an abelian integral on the
Riemann surface associated with M (q) and the quantities λ∗ (a, b, α, β, γ, δ) are
‘periods’. Since the expression u2 − u + 1/6 does not vanish for rational u,
fa,b (z) is an abelian integral strictly of the third kind. It might be an interesting
problem to determine the ‘periods’, λ∗ (a, b, α, β, γ, δ).
We had defined the function ϑ1 (w, z) by the infinite product (36); now, we
shall identify it with the infinite series expansion by which it is usually defined.
For this purpose, we define for z ∈ H and complex w,
f (w, z) =
∞
X
n=−∞
1 2
1
1
eπi(n+ 2 ) z+2πi(n+ 2 )(w− 2 ) .
(46)
37
Kronecker’s Limit Formulas
33
Clearly f (w, z) is a regular function of w and z. Moreover, it is easy to see that
f (w + 1, z) = − f (w, z).
(47)
Also replacing n by n + 1 on the right-hand side of (46), we have
f (w, z) =
∞
X
n=−∞
2πiw+πiz
= −e
1 2
eπiz(n+1+ 2 )
+2πi(n+1+ 12 )(w− 21 )
f (w + z, z),
i.e.
f (w + z, z) = −e−2πiw−πiz f (w, z).
(48)
We know already that
ϑ1 (w + 1, z) = −ϑ1 (w, z)
and
ϑ1 (w + z, z) = −e−2πiw−πiz ϑ1 (w, z).
f (w, z)
is a doubly periodic
ϑ1 (w, z)
function of w with independent periods 1 and z and is regular in w except
possibly when w = m + nz with integral m and n, since ϑ1 (w, z) has simple
zeros at these points. But f (w, z) also has zeros at these points, since, replacing
n by −n − 1 on the right-hand side of (46), we have f (−w, z) = − f (w, z) and
hence f (0, z) = 0 and from (47) and (48), f (m + nz, z) = 0 for integral m and
n. Thus h(w, z) is a doubly periodic entire function of w and by Liouville’s
theorem, it is independent of w, i.e. h(w, z) = h(z).
The function h(z) is a regular function of z in H and to determine its behaviour under the modular transformations, it suffices to consider its behaviour
under the transformations z → z + 1 and z → −z−1 . We can show easily that
h(z + 1) = h(z). Moreover, by (42),
r
!
z
w 1
πiw2 /z 1
,− = e
ϑ1 (w, z).
ϑ1
z z
i
i
Hence,for fixed z ∈ H, the function h(w, z) =
Let us assume, for the present, that we have proved the formula
r
!
w 1
z
2 1
, − = eπiw .z
f (w, z).
f
z z
i
i
(49)
It will then follow that h(−1/z) = h(z). Hence h(z) is a modular function and it
is indeed regular everywhere in H.
38
Kronecker’s Limit Formulas
34
Now, a fundamental domain of the elliptic modular group in H is given by
the set of z = x + iy ∈ H for which |z| ≥ 1 and −1/2 ≤ x ≤ 1/2. Let z tend
to infinity in this fundamental domain. Then it is easy to see that the functions
ϑ1 (w, z)e−πiz/4 and f (w, z)e−πiz/4 tend to the same limit −i(eπiw − e−πiw ). Hence
h(z) − 1 tends to zero as z tends to infinity in the fundamental domain. But
then, being regular in H and invariant under modular transformations, it attains
its maximum at some point in H. By the maximum modulus principle, h(z) − 1
is a constant and since h(z) − 1 tends to zero as z tends to infinity, h(z) = 1. In
other words, ϑ1 (w, z) = f (w, z).
To complete the proof, all we need to do is to prove (49). We may first
rewrite f (w, z) as
1 2
f (w, z) = e−πi(w− 2 ) /z
∞
X
1
2
eπiz(n+ 2 +w/z−1/2z)
n=−∞
2
Let ξ > 0 and let g(x) = e−πξx , for −∞ < x < ∞. Then the series
∞
P
39
g(x+
n=−∞
n) converges uniformly in the interval 0 ≤ x ≤ 1 and by the Poisson summation
formula (7),
Z ∞
∞
∞
X
X
2
2πinx
−πξ(x+n)2
e
e−πξx −2πinx dx,
(50)
e
=
−∞
n=−∞
n=−∞
(provided the series on the right-hand side converges). Now
Z
∞
2
e−πξx
2
−2πinx
−∞
dx =
e−πn /ξ
√
ξ
Z
∞
e−π(x+in/
−∞
√
ξ)2
dx,
p
( ξ > 0).
By the Cauchy integral theorem, one can show that
Z ∞
Z ∞
√
2
−π(x+in/ ξ)2
dx =
e−πx dx = γ(say).
e
−∞
−∞
It is now obvious that the series on the right-hand side of (50) converges absolutely. Hence
∞
∞
X
γ X −πn2 /ξ+2πinx
2
e−πξ(x+n) = √
e
.
(51)
ξ n=−∞
n=−∞
Setting ξ = 1 and x = 0 in (51), we have immediately γ = 1. Both sides of
(51) represent, for complex x, analytic functions of x and since they are equal
for real x, formula (51) is true even for complex x. Moreover, both sides of
(51) represent analytic functions of ξ, for complex ξ with Re ξ > 0. Since they
Kronecker’s Limit Formulas
35
coincide for real ξ > 0, we see again that (51) is valid for all complex ξ with
Re ξ > 0. We have thus the theta-transformation formula
∞
X
∞
1 X −πn2 /ξ+2πinv
2
e−πξ(n+v) = √
e
ξ n=−∞
n=−∞
(52)
√
valid for all complex v and complex ξ with Re ξ > 0, ξ denoting that branch
which is positive for real ξ > 0.
Settingξ = −iz and v = (1/2) + (w/z) − (1/2z) in (53), we have
f (w, z) =
1 2
∞
e−πi(w− 2 ) /z X −πin2 /z+2πin((1/2)+(w/z)−(1/2z))
e
.
q
z
i
n=−∞
Now
2
e−πin
/z+2πin((1/2)+(w/z)−(1/2z))−πi(w− 12 )2 /z
1
1 2
= e−πi(n+ 2 ) /z+2πi(n+ 2 )(w/z−1/2)−πiw
Thus
ie−πiw
f (w, z) = q
2
/z
z
i
2
/z+πi/2
.
!
w 1
,− ,
f
z z
which is precisely (49). Thus
Proposition 5. For the elliptic theta-function ϑ1 (w, z) defined by the infinite
product (36), we have the series-expansion
ϑ1 (w, z) =
∞
X
1 2
1
1
eπi(n+ 2 ) z+2πi(n+ 2 )(w− 2 ) .
n=−∞
Using the method employed above, we shall also prove
Proposition 6. The Dedekind η-function η(z) has the infinite series expansion
η(z) = eπiz/12
∞
X
(−1)λ eπiz(3λ
∞
X
(−1)∗ eπiz(3λ
2
−λ)
λ=−∞
Proof. Let us define for z ∈ H,
f (z) = eπiz/12
λ=−∞
2
−λ)
.
40
Kronecker’s Limit Formulas
36
The function f (z) is regular in H and moreover, it is clear that f (z + 1) =
f (z)eπi/12 . Again,
πiz/12
f (z) = e
∞
X
1
2
e2πiz(λ− 6 (1−1/z))
−πiz(1−1/z)2 /12
λ=−∞
= e−πi/12z eπi/6
∞
X
1
2
e3πiz(λ− 6 (1−1/z))
λ=−∞
−πi/12z πi/6
=e
e
r
∞
i X −πiλ2 /3z−πiλ(1−1/z)/3
e
,
3z λ=−∞
√
using formula (52). Here i/3z denotes that branch which is positive for z = i.
∞
P
2
Now we split up
e−πi/3zλ −πiλ/3(1−1/z) as g0 (z) + g1 (z) + g2 (z), where for
λ=−∞
k = 0, 1, 2,
gk (z) =
∞
X
2
e−πi(3µ+k)
/3z−πi(3µ+k)(1−1/z)/3
.
n=−∞
Clearly,
g0 (z) =
∞
X
(−1)µ e−πi(3µ
2
−µ)/z
,
µ=−∞
g1 (z) = e−πi/3 g0 (z),
∞
X
2
e−πi(9µ +12µ+4)/3z−πi(3µ+2)(1−1/z)/3
g2 (z) =
µ=−∞
= e−2πi/3z−2πi/3
∞
X
(−1)µ e−3πiµ(µ+1)/z .
µ=−∞
Now
∞
X
(−1)µ e−3πiµ(µ+1)/z = −
µ=−∞
∞
X
(−1)µ e−3πiµ(µ+1)/z ,
µ=−∞
on replacing µ by −1 − µ, and hence g2 (z) = 0. Thus
r
∞
X
i πi/6
2
−πi/12z
f (z) = e
(e + e−πi/6 )
(−1)µ e−πi(3µ −µ)/z
3z
µ=−∞
r
!
i
1
f − .
=
z
z
41
Kronecker’s Limit Formulas
37
Asa consequence, the function h(z) = f (z)/n(z) is invariant under the transformations z → z + 1 and z → −1/z and hence under all modular transformations.
Moreover, since η(z) does not vanish anywhere in H, h(z) is regular in H; further, as z tends to infinity in the fundamental domain, h(z) − 1 tends to zero.
Now by the same argument as for ϑ1 (w, z) above, we can conclude that h(z) = 1
and the proposition is proved.
The integers (1/2)(3λ2 − λ) for λ = 1, 2, . . . are the so-called ‘pentagonal
numbers’.
We now give another interesting application of Kronecker’s second limit
formula. Let us consider, once again, formula (37). The left-hand side may be
looked upon as a trigonometric series in u and v; it is true, of course, that u and
v are not entirely independent. We shall now see how, using the infinite series
expansion of ϑ1 (w, z), Kronecker showed that the function
2
ϑ1 (w, z)ϑ1 (q, −r)e(πi(w−q)
)/(z−r)
has a valid Fourier expansion in u and v and derived a beautiful formula from
(37). First,
ϑ1 (w, z) =
ϑ1 (q, −r) =
∞
X
n=−∞
∞
X
1 2
1
1
eπiz(n+ 2 ) +2πi(n+ 2 )(w− 2 ),
1 2
1
1
1 2
1
1
e−πir(m+ 2 ) +2πi(m+ 2 )(q− 2 ) ,
m=−∞
replacing m by −m − 1, we have
ϑ1 (q, −r) =
∞
X
e−πir(m+ 2 ) −2πi(m+ 2 )(q− 2 ) .
m=−∞
Hence
ϑ1 (w, z)ϑ1 (q, −r)
∞
∞
X
X
1
1
1 2
1
1
1 2
=
eπi{z(n+ 2 ) +2(n+ 2 )(w− 2 )−r(m+ 2 ) −2(m+ 2 )(q− 2 )} .
m=−∞ n=−∞
This double series converges absolutely and replacing n by n + m in the inner
sum, we have
42
Kronecker’s Limit Formulas
38
ϑ1 (w, z)ϑ1 (q, −r)
∞
∞
X
X
1 2
1
1
2
eπi(z−r)(m+ 2 ) +2πi(m+ 2 )(w+nz−q)+πin z+2πin(w− 2 )
=
=
m=−∞ n=−∞
∞
X
n πin2 z+2πinw−πi(w+nz−q)2 /(z−r)
(−1) e
n=−∞
∞
X
1
2
eπi(z−r)(m+ 2 +(w+nz−q)/(z−r))
m=−∞
Applying the theta-transformation formula to the inner sum, we have
43
2
ϑ1 (w, z)ϑ1 (q, −r)eπi(w−q) /(z−r)
r
∞
i X
2
2 2
(−1)n eπin z+2πinw−πi(n z +2nz(w−q))/(x−r) ×
=
z − r n=−∞
×
=
∞
X
2
(−1)m e−πim
/(z−r)−2πim(w−q+nz)/(z−r)
m=−∞
r
i X
(−1)mn+m+ne −πi(m+nz)(m+nr)/(z−r)+2πi((m+nz)q−(m+nr)w)/(z−r)
z − r m,n
√
i/z − r being that branch which assumes the value 1 for z − r = i.
Let
i
Q(ξ, η) =
(ξ + ηz)(ξ + ηr) = aξ2 + bξη + cηn ,
z−r
where
i
i(z + r)
izr
a=
,b =
and c =
.
z−r
z−r
z−r
Then
2
ϑ1 (w, z)ϑ1 (q, −r)eπi(w−q) /(z−r)
r
i X
(−1)mn+m+n e−πQ(m,n)+2πi((m+nz)q−(m+nr)w)/(z−r)
=
z − r m,n
Formula (37) now becomes
1 X′ e2πi(mu+nv)
−
= log
2π m,n Q(m, n)
q
i
z−r
P
m,n (−1)
mn+m+n −πQ(m,n)+2πi(mu+nv)
e
η(z)η(−r)
It is remarkable that the quadratic form Q(m, n) emerges undisturbed on the
44
Kronecker’s Limit Formulas
39
right-hand
side; the right-hand side is free from z and r, except for the fac√
tors i/(z − r) and η(z)η(−r). To eliminate z and r therefrom, we differentiate
2
ϑ1 (w, z)ϑ1 (q, −r)eπi(w−q) /(z−r) once with respect to w and q at w = 0, q = 0, and
then we get
r
i X
′
′
(−1)mn+m+n e−πQ(m,n)×
ϑ1 (0, z)ϑ1 (0, −r) =
z − r m,n
2πi(m + nz) −2πi(m + nr)
·
,
z−r
z−r
i.e.
i.e.
3
r

i  X

(−1)(m+1)(n+1) Q(m, n)e−πQ(m,n) ,
η3 (z)η3 (−r) = 
z − r m,n
η(z)η(−r) =
r
i
z−r
 31





X
(m+1)(n+1)
−πQ(m,n) 
(−1)
Q(m, n)e






m,n
the branch of the cube root on the right-hand side being determined by the
condition that it is real and positive when r = z, since η(z)η(−z) > 0.
We have then the following formula due to Kronecker, namely
P
mn+m+n −πQ(m,n)+2πi(mu+nv)
e
1 X′ e2πi(mu+nv)
m,n (−1)
−
(53)
= log n
o1 .
P
2π m,n Q(m, n)
(m+1)(n+1) Q(m, n)e−πQ(m,n) 3
(−1)
m,n
It is remarkable that, eventually, on the right-hand side of (53), z and r do
not appear explicitly and only the quadratic form Q(m, n) appears on the right.
The quadratic form Q(ξ, n) = dξ2 +bξη+cη2 introduced above is a complex
quadratic form with discriminant
b2 − 4ac =
−(r + z)2 + 4rz
= −1.
(z − r)2
Moreover,its real part is positive-definite for real
iξ and η. For proving
this,
we have to show that for (ξ, n) , (0, 0), Re
(ξ + zη) · (ξ + rη) > 0,
z−r
1
i.e. Im − z−r (ξ + zn)(ξ + rη) > 0. If η = 0, this is trivial. If η , 0, we may
take η = 1 without loss of generality and then we have only to show that
!
!
1
1
z−r
−
> 0, i.e., Im
> 0.
Im
(ξ + z)(ξ + r)
ξ+r ξ−z
45
Kronecker’s Limit Formulas
40
But this is obvious, since z, −r ∈ H and ξ being real, ξ + z, −(ξ + r) and hence,
−1/(ξ + z), 1/(ξ + r) are all in H.
Conversely, if Q(ξ, η) = aξ2 + bξη + cη2 is a complex quadratic form with
real part positive-definite for real ξ and η and discriminant −1 and if Re a > 0,
i
it can be written as
(ξ + ηz)(ξ + ηr) for some z, −r ∈ H. In fact, if Re a > 0,
z−r
Q(ξ, η) = a(ξ + ηz)(ξ + ηr), where z, r are the roots of the quadratic equation
aλ2 − bλ + c = 0. Since the real part of Q(ξ, η) is positive definite, z and r
have both got to be complex. Now the discriminant of Q(ξ, η) is −1 and hence
a2 (z−r)2 = −1, i.e. a = ±i/(z−r). We may take a = i/(z−r) and then z−r ∈ H.
i
Now Q(ξ, η) =
(ξ + ηz)(ξ + ηr). We have only to show that z, −r ∈ H.
z−r
i
Consider the expression Q(ξ, 1) =
(ξ + z)(ξ + r). If both z and r are in H
z−r
then as ξ varies from −∞ to +∞, the argument of Q(ξ, 1) decreases by 2π and
if both −z and −r are in H then as ξ varies from −∞ to +∞, the argument of
Q(ξ, 1) increases by 2π. But neither case is possible, since Re Q(ξ, 1) > 0 and
so Q(ξ, 1) lies always in the right half-plane for real ξ. Hence either z, −r or
−z, r ∈ H. But, again, since z − r ∈ H we see that z, −r ∈ H.
When we subject the quadratic form Q(ξ, η) to the unimodular transformation (ξ, η) → (αξ + βη, γξ + δη) where α, β, γ, δ are integers such that
αδ − βγ = ±1, then Q(ξ, η) goes over into the quadratic form Q∗ (ξ, η) =
Q(αξ + βη, γξ + δη). The quadratic forms Q∗ (ξ, η) and Q(ξ, n) are said to be
equivalent. Q∗ (ξ, η) again has discriminant −1 and its real part is positivedefinite for real ξ and η.
Let u∗ = αu + γv and v∗ = βu + δv. Then we assert that the right-hand
side of (53) is invariant if we replace Q(m, n), u and v respectively byQ∗ (m, n),
u∗ and v∗ . This is easy to prove, for the series on the right-hand side converge
absolutely and when m, n run over all integers independently, then so do m∗ =
∗
∗
αm + βn, n∗ = γm + δn. And all we need to verify is that (−1)(m +1)(n +1) =
(m+1)(n+1)
(m+1)(n+1)
(−1)
. This is very simple to prove; for (−1)
= +1 unless m
and n are both even; and m and n are both even if and only if m∗ and n∗ are
both even.
The invariance of the left hand side of (53) under the transformation Q(m, n),
u, v to Q∗ (m, n), u∗ , v∗ could be also directly proved by observing that for
P e2πi(mu+nv)
is invariant under this transformation and hence, by
Re s > 1, ′
(Q)(m, n) s
analytic continuation, the invariance holds good even for s = 1.
46
Kronecker’s Limit Formulas
41
5 The Epstein Zeta-function
We shall consider here a generalization of the function ζQ (s, u, v) introduced in
§ 3. Let Q be the matrix of a positive-definite quadratic form Q(x1 , . . . , xn ) in
x1
the n variables x1 , . . . , xn (n ≥ 2). Further let x denote the n-rowed column ... .
xn
A homogeneous polynomial P(x) = P(x1 , . . . , xn ) of degree g in x1 , . . . , xn
is called a spherical function (Kugel-funktion) of order g with respect to
Q(x1 , . . . , xn ), if it satisfies the differential equation
X
q∗i j
1≤i, j≤n
∂2 P(x)
= 0,
∂xi ∂x j
where the matrix (q∗i j ) = Q−1 .
Let, for a matrix A, A′ denote its transpose; we shall abbreviate A′ QA as
Q[A], for an n-rowed matrix A. By an isotropic vector of Q, we mean a complex column w satisfying Q[w] = 0. It is easily verified that if w is an isotropic
vector of Q, then the polynomial (x′ Qw)g is a spherical function of order g
with respect to Q(x1 , . . . , xn ). Moreover, it is known thatif P(x) is a spherical
M
P
(x′ Qwm )g ,
function of order g with respect to Q(x1 , . . . , xn ), then P(x) =
m=1
w1 , . . . , w M being isotropic vectors of Q.
Let u, v be two arbitrary n-rowed real columns and g, a non-negative integer. Further, let P(x) be a spherical function of order g with respect to
Q(x1 , . . . , xn ). We define, for σ > n/2, the zeta-function
ζ(s, u, v, Q, P) =
X
m+v,0
′
e2πim u
P(m + v)
(Q[m + v]) s+g/2
where m runs over all n-rowed integral columns such that m + v , 0 (0 being
the n-rowed zero-column). By the remark on P(x) above, ζ(s, u, v, Q, P) is a
linear combination of the series of the form
X
m+v,0
′
e2πim u
((m + v)′ Qw)g
,
(Q[m + v]) s+g/2
w being an isotropic vector of Q. These series have been investigated in detail by Epstein; in special cases as when g = 0 and Q is diagonal, Lerch has
considered them independently and has derived for them a functional equation
which he refers to as the “generalized Malmsten-Lipschitz relation.”
47
Kronecker’s Limit Formulas
42
The series introduced above converge absolutely for σ > n/2, uniformly in
every half-plane σ ≥ n/2 + ǫ(ǫ > 0), as can be seen from the fact that they have
a majorant of the form
!g
n
P
|m
+
v
|
i
i
X
i=1
c·
!σ+g/2
n
m+v,0 P
|mi + vi |2
i=1
for a constant c depending only on Q, w and g. Hence they represent analytic
functions of s for σ > n/2. Thus ζ(s, u, v, Q, P) is an analytic function of s
for σ > n/2.
We shall study the analytic continuation and the functional equation of
ζ(s, u, v, Q, P). The proof will be based on Riemann’s method of obtaining the
functional equation of the ζ-function using the theta-transformation formula.
First we obtain the following generalization of (52).
Proposition 7. If Q is the matrix of a positive-definite quadratic form in m
variables with real coefficients and v an n-rowed complex column, then
X
X
1
′
−1
e−πq[m+v] = |Q|− 2
e−πQ |m|+2πm v ,
(54)
m
m
1
where, on both sides m runs over all n-rowed integral columns and |Q| 2 is the
positive square root of |Q|, the determinant of Q.
Proof. We shall assume formula (54) proved for Q and v of at most n − 1 rows
and uphold it for n. For n = 1, the formula has been proved already.
Now, it is well known that
#
!"
!
P q
E P−1 q
P
0
(55)
A= ′
= ′
q r
0 r − P−1 [q] 0′
1
where P is (n − 1)-rowed and symmetric and E, the (n − 1)-rowed
identity
matrix. Setting λ = r − P−1 [q] and writing m = 1k and v = wu with k and u of
n − 1 rows, we have, in view of (55),
X
X
X
−1
2
E −πλ(l+w)
e−πQ[m+v] =
e−πP[k+u+P q(l+w)] ,
m
1
k
m, k running over all n-rowed and (n − 1)-rowed integral columns respectively
and l over all integers. By induction hypothesis,
X
X
′
−1
−1
−1
1
e−πP[k+u+P q(l+w)] = |P|− 2
e−πP [k]+2πik (u+P q(l+w))
k
k
48
Kronecker’s Limit Formulas
43
Thus
X
m
1
e−πQ[m+v] = |P|− 2
X
−1
e−πP [k] + 2πik′ u
k
X
′ −1 (l+w)
2
e−πλ(l+w) +2πik P
q
l
Now, by (52),
49
∞
X
′ −1
2
e−πλ(l+w) +2πik P
q(l+w)
l=−∞
= e−πλw
2
+2πik′ P−1 qw
∞
X
e−πλl
2
−2πil(k′ P−1 q+iλw)
l=−∞
− 21
= λ e−πλw
2
′ −1 w
+2πik P q
∞
X
e−πλ
−1
(l−k′ P−1 q−iλω)2
l=−∞
1
= λ− 2
∞
X
e−πλ
−1
′ −1
(l−k P q)2 +2πilw
l=−∞
But |Q| = |P|λ and Q−1 [m] = P−1 [k] + λ−1 (−k′ P−1 q + 1)2 and hence
X
m
1
e−πQ[m+v] = |Q|− 2
X
e−πQ
−1
|m|+2πi(k′ u+lw)
k,l
and (54) is proved.
Formula (54) can also be upheld by using the Poisson summation formula
in n variables. Further, it can be shown to be valid even for complex symmetric
Q whose real part is positive (i.e. the matrix of a positive-definite quadratic
form). For, Q−1 again has positive real part and the absolute convergence of
the series on both sides is ensured. Moreover, considered as functions of the
n(n + 1)/2 independent elements of Q, they represent analytic functions and
since they are equal for all real positive Q, we see, by analytic continuation,
that they are equal for all complex symmetric Q with positive real part.
Let u be another arbitrary complex n-rowed column. Replacing v by v −
iQ−1 u in (54), we obtain
X
X
1
′
−1
−1
−1
e−πQ[m+v−iQ u] = |Q|− 2
e−πQ [m]+2πim (v−iQ u) ,
m
i.e.
X
m
m
′
1
e−πQ[m+v]+2πim u = |Q|− 2
X
m
e−πQ
−1
[m−u]+2πi(m−u)′ v
.
(56)
Kronecker’s Limit Formulas
44
Let w be an isotropic vector of Q and λ, an arbitrary complex number. Replacingv by v + λw in (56), we have
X
′
′
e−πQ[m+v]−2πλ(m+v) Qw+2πim u
m
1
= |Q|− 2
X
e−πQ
−1
[m−u]+2πi(m−u)′ v+2πiλ(m−u)′ w
.
m
Both sides represent analytic functions of λ and differentiating them both, g
times with respect to λ at λ = 0, we get
X
′
e−πQ[m+v]+2πim u ((m + v)′ Qw)g
m
′
e−2πiu v − 1 X −πQ−1 [m−u]+2πim′ v
e
((m − u)′ w)g .
|Q| 2
ig
m
=
(57)
Associated with a spherical function P(x) with respect to Q[x], let us define P ∗ (x) = P(Q−1 x); P ∗ (x) is a spherical function of order g with respect
m
P
to Q−1 [x]. Moreover P ∗∗ (x) = P(x). Further, if P(x) = (x′ Qwi )g , then
P ∗ (x) =
m
P
i=1
i=1
(x′ wi )g . Thus, if we set
f (Q, u, v, P) =
X
′
e−πQ[m+v]+2πim u P(m + v),
m
where m runs over all n-rowed integral columns, then we see at once from (57)
that f (Q, u, v, P) satisfies the functional equation
′
1
ig e2πiu v f (Q, u, v, P) = |Q|− 2 f (Q−1 , v, −u, P ∗ ).
If now we replace Q by xQ for x > 0, then P(x) goes into xg P(x) and
P (x) remains unchanged and we have
∗
′
1
ig e2πiu v f (xQ, u, v, P) = x−n/2−g |Q|− 2 f (x−1 Q−1 , v, −u, P ∗ ).
(58)
Thus we have
Proposition 8. Let Q be an m-rowed complex symmetric matrix with positive
real part, P(x) a spherical function of order g with respect to Q and P ∗ (x) −
50
Kronecker’s Limit Formulas
45
P(Q−1 x). Let u and v be two arbitrary m-rowed complex columns and x > 0.
Then
X
′
′
e−πxQ[m+v]+2πim u P(m + v)
e2πiu vi g
m
=x
−n/2−g
1
|Q|− 2
X
e−πx
−1
Q−1 [m−u]+πim′ v
m
P ∗ (m − u).
51
To study the analytic continuation of ζ(s, u, v, Q, P), we obtain an integral
representation of the same, by using the well-known formula due to Euler,
namely, for t > 0 and Re s > 0,
Z ∞
dx
−s
−s
π Γ(s)t =
x s e−πtx .
x
0
For σ > n/2, we then have
g
π−(s+g/2) Γ s + ζ(s, u, v, Q, P)
2
Z ∞
X
dx
2πim′ u
e
P(m + v)
x s+g/2 e−πxQ[m+v]
=
x
0
m+v,0


Z ∞




X


′
 dx

−πxQ[m+v]+2πim u
e
P(m
+
v)
=
x s+g/2 
,





 x
m+v,0
0
(59)
in view of the absolute convergence of the series, uniform for σ ≥ n/2 + ǫ(ǫ >
0).
Let now, for an n-rowed real column x,



0, if x is not integral,




ρ(x, g) = 
1, if x is integral and g = 0,




0, if x is integral and g > 0.
Then
X
m+v,0
′
′
e−πxQ[m+v]+2πim u P(m + v) = f (xQ, u, v, P) − e−2πiu v ρ(v, g).
From (59), we see, as a consequence, that
g
π−(s+g/2) Γ s + ζ(s, u, v, Q, P)
2
Kronecker’s Limit Formulas
46






X


′
 dx

−πxQ[m+v]+2πim u
s+g/2
e
P(m + v)
=
x
+





 x
m+v,0
1
Z 1
Z 1
dx
dx
′
x s+g/2 .
+
x s+g/2 f (xQ, u, v, P) − ρ(v, g)e−2πiu v
x
x
0
0
Z
∞
In view of (58),
Z 1
52
x s+g/2 f (xQ, u, v, P)
0
1
=
|Q|− 2
′
g
i e2πiu v
Z
0
dx
x
1
x s−g/2−n/2 f (x−1 Q−1 , v, −u, P ∗ )
dx
.
x
Again, since
f (x−1 Q−1 , v, −u, P ∗ )
X
−1 −1
′
′
=
e−πx Q [m−u]+2πim v P ∗ (m − u) + ρ(u, g)e2πiu v ,
m−u,0
we have
Z
1
dx
x s−g/2−n/2 f (x−1 Q−1 , v, −u, P ∗ )
x
0


Z 1




X


−1 −1
′
 dx

−πx Q [m−u]+2πim v
∗
s−g/2−n/2
e
P (m − u)
=
x
+





 x
m−u,0
0
Z 1
dx
′
+ ρ(u, g)e2πiu v
x s−(g/2)−(n/2)
x
0


Z ∞




X


−1
′
 dx

−πxQ [m−u]+2πim v
∗
e
P
(m
−
u)
=
x(n/2)−s+(g/2) 
+





 x
m−u,0
1
′
ρ(u, g)e2πiu v
+
g n .
s− −
2 2
Thus, for σ > n/2,
g
π−(s+g/2) Γ s + ζ(s, u, v, Q, P)
2
′
− 12
ρ(v, g)e−2πiu v
|Q| ρ(u, g)
+
= g
g n −
s+
ig s − −
2
2 2
Kronecker’s Limit Formulas
47


Z


 ∞


X


′
 dx


−πxQ[m+v]+2πim u
s+g/2

e
P(m + v)
+ 
x
+





 x
m+v,0
1

 
Z ∞
1





X


|Q|− 2
−1
′

 dx 
−πxQ [m−u]+2πim v
∗
x(n/2)−s+(g/2) 
e
×
P
(m
−
u)
+ g 2πiu′ v
 .





ie
m−u,0
 x
1
(60)
53
One verifies easily that the functions within the square brackets on the
π s+g/2
is also an enright-hand side of (60) are entire functions of s. Since
Γ(s + g/2)
tire function of s, formula (60) gives the analytic continuation of ζ(s, u, v, Q, P)
into the whole s-plane. The only possible singularities arise from those of




−2πiu′ v 
− 12


s+g/2 


g)e
ρ(v,
ρ(u,
|Q|
g)
π


−



g
g
g 
n


g


i s − 2 − 2

s+
Γ s+
2
2
Now
π s+g/2
π s+g/2
=
g
Γ s + 2g + 1
Γ s + g2 s +
2
is an entire function of s and so s = −g/2 cannot be a singularity of ζ(s, u, v, Q, P).
Moreover, if g > 0 or if u is not integral, then ρ(u, g) = 0 and hence in
these cases, s = (g + n)/2 can not be a pole of ζ(s, u, v, Q, P). If g = 0,
and u is integral, ζ(s, u, v, Q, P) has a simple pole at s = n/2 with residue
1
πn/2 |Q|− 2 /Γ(n/2). Moreover, from (60) it is easy to verify that ζ(s, u, v, Q, P)
satisfies the functional equation
g
π−sΓ s + ζ(s, u, v, Q, P)
2
′
e−2πiu v − 1 −(n/2−s) n
g n
−1
∗
2π
=
−u,
Q
,
P
(61)
|Q|
Γ
−
s
+
−
s,
v,
ζ
ig
2
2
2
Wehave then finally.
Theorem 3. The function ζ(s, u, v, Q, P) has an analytic continuation into the
whole s-plane, which is an entire function of s if either g > 0 or if g = 0 and u
is not integral. If g = 0 and u is integral, then ζ(s, u, v, Q, P) is meromorphic
in the entire s-plane with the only singularity at s = n/2 where it has a simple
1
pole with the residue πn/2 /(|Q| 2 Γ(n/2)). In all cases, ζ(s, u, v, Q, P) satisfies
the functional equation (61).
54
Kronecker’s Limit Formulas
48
−1
2
Let now
notation.
Q(x
x n = 2m and
u1 , x 2 ) = y |xv 1 + x2 z| ,−zin our earlier
1
1
2
1
are
isotropic
Let x = x2 , m = m2 , u = −u1 and v = v2 . Both 1 and −z
1
vectors of Q[x] but we shall take the spherical function P(x) = (−i(x1 + x2 z))g ,
corresponding to the former. Then by the above, the function
ζ(s, u, v, Q, P) =
y s+g/2 X 2πi(m1 u2 −m2 u1 ) (m1 + v1 + z(m2 + v2 ))g
e
ig m+v,0
|m1 + v1 + z(m2 + v2 )|2s+g
satisfies the functional equation
g
π−s Γ s + ζ(s, u, v, Q, P)
2
′
e−2πiu v −(1−s) g
=
π
Γ
1
−
s
+
ζ(1 − s, v, −u, Q−1 , P ∗ ).
ig
2
Now Q−1 [x] has the simple form y−1 |x1 z − x2 |2 and moreover, P ∗ (x) = (−x1 z +
x2 )g . Thus for σ > 1,
ζ(s, v, −u, Q−1 , P ∗ )
X
(−(m1 − u2 )z + m2 + u1 )g
= y s+g/2
e2πi(m1 v1 +m2 v2 )
| − z(m1 − u2 ) + m2 + u1 |2s+g
m−u,0
X
((m1 + u1 ) + z(m2 + u2 ))g
.
= y s+g/2
e2πi(m1 v2 −m2 v1 )
|(m1 + u1 ) + z(m2 + u2 )|2s+g
(mm12 )+(uu12 ),0
Let us now define for u∗ =
ζ(s, u∗ , v∗ , z, g) = y s
X
u v ∗
1
1
u2 , v = v2 , the function
m+v∗ ,0
e2πi(m1 u2 −m2 u1 )
(m1 + v1 + z(m2 + v2 ))g
,
|m1 + v1 + z(m2 + v2 )|2s+g
forσ > 1. It is clear that ζ(s, u∗ , v∗ , z, g) = ig y−g/2 ζ(s, u, v, Q, P) and from
(??) we see that ζ(s, v∗ , u∗ , z, g) = y−g/2 ζ(s, v, −u, Q−1 , P ∗ ). If now we defined
ϕ(s, u∗ , v∗ , z, g) = π−s Γ(s + g/2)ζ(s, u∗ , v∗ , z, g), we deduce from above that
ϕ(s, u∗ , v∗ , z, g) satisfies the nice functional equation
ϕ(s, u∗ , v∗ , z, g) = e2πi(u1 v2 −u2 v1 ) ϕ(1 − s, v∗ , u∗ , z, g).
In the case g > 0, ζ(s, u∗ , v∗ , z, g) is an entire function of s and one can
ask for analogues of Kronecker’s second limit formula even here. For even g,
one gets by applying the Poisson summation formula, a limit formula which is
connected with elliptic functions. For odd g, the limit formulas which one gets
55
Kronecker’s Limit Formulas
49
are more complicated and involve Bessel functions. If we take the particular
case g = 1, there occurs in the work of Hecke, a limit formula (as s tends to 1/2)
which has an interesting connection with the theory of complex multiplication.
For n > 2, Epstein has obtained for u = 0, v = 0 and g = 0, an analogue of
the first limit formula of Kronecker. This formula involves more complicated
functions that the Dedekind η-function. Perhaps, in general for n > 2, one
cannot expect to get a limit formula which would involve analytic functions of
several complex variables.
Bibliography
[1] S. Bochner: Vorlesungen über Fouriersche Integrale, Chelsea, New York,
(1948).
[2] R. Dedekind: Erläuterungen zu den Fragmenten XXVIII, Collected Works
of Bernhard Riemann, Dover, New York, (1953), 466-478.
[3] P. Epstein: Zur Theorie allgemeiner Zeta funktionen, I, II, Math, Ann. 56
(1903), 615-644, ibid, 63 (1907), 205-216.
[4] E. Hecke: Bestimmung der Perioden gewisser Integrale durch die Theorie
der Klassenkorper, Werke. Gottingen (1959), 505-524.
[5] F. Klein: Uber die elliptischen Normalkurven der n-ten Ordnung,
Gesamm. Math. Abhandlungen. Bd. III. Berlin. (1923). 198-254.
[6] Kronecker:Zur Theorie der Clliptischen Funktionen, Werke. Vol. 4. 347495, vol. 5, 1-132, Leipzig, (1929).
[7] M. Lerch: Zakladové Theorie Malmstenovskych rad, Rozpravy česke
akademie Cišare Frantiska Josefa, Bd. 1, No. 27, (1892), 525-592.
[8] C. Meyer: Die Berechnung der Klassenzahl abelscher Körper über
quadratischen Zahlkörpern, Akademie Verlag, Berlin, 1957.
[9] H. Rademacher: Zur Theorie der Modulfunktionen, J. für die reine u.
angew. Math., vol. 167 (1931), 312-336. Collected papers.
[10] B. Schoenesberg: Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523.
r
!
τ
1
= η(τ)
, Mathematika 1
[11] C. L. Siegel: A simple proof of η −
η
i
(1954), 4. Ges Abhand.
50
56
Bibliography
51
[12] E. T. Whittaker and G. N. Watson: Course of Modern Analysis, Cambridge, (1946).
Chapter 2
Applications of Kronecker’s
Limit Formulas to Algebraic
Number Theory
1 Kronecker’s solution of Pell’s equation
57
Let K be an algebraic number field of degree n over Q, the field of rational
numbers. Let, for any ideal a in K, N(a) denote its norm. Further let s = σ + it
be a complex variable. Then for σ > 1, we define after Dedekind, the zeta
function
X
(N(a))−s ,
ζK (s) =
a
where the summation is over all non-zero integral ideals of K.
More generally, with a character χ of the ideal class group of K, we associate the L-series
Y
X
(1 − χ(p)(N(p))−s )−1 ,
χ(a)(N(a))−s =
LK (s, χ) =
p
a
for σ > 1. The product on the right runs over all prime ideals p of K.
We have h such series associated with all the h characters of the ideal class
group of K. It has been proved by Hecke that these functions LK (s, χ) can
be continued analytically into the whole plane and they satisfy a functional
equation for s → 1 − s.
52
Applications to Algebraic Number Theory
53
Now, χ being an ideal class character, we may write
X
X
LK (s, χ) =
χ(A) (N(a))−s ,
a∈A
A
A running over all the ideal classes of K. If we define ζ(s, A) =
P
(N(a))−s ,
a∈A
then it can be shown that lim(s − 1)ζ(s, A) = κ, a positive constant depending
s→1
only upon the field K and not upon the ideal class A. (For example,
in the
√
case of an imaginary quadratic field of discriminant d,√ κ = 2/w −d, w being
the number of roots of unity in K and κ = 2 log ǫ/ d in the case of a real
quadratic field of discriminant d, where ǫ is the fundamental unit and ǫ > 1).
P
Since ζK (s) = ζ(s, A), we have
A
lim(s − 1)ζK (s) =
s→1
X
A
lim(s − 1)ζ(s, A) = κ · h
s→1
where h denotes the class number of K.
From now on, we shall be concerned only with quadratic fields of discriminant d.
We have then, on direct computation,
ζK (s) = ζ(s)Ld (s),
(63)
where ζ(s) is the Riemann zeta-function and Ld (s) is defined as follows:
!
∞
X
d −s
Ld (s) =
n ,
n
n=1
(σ > 1)
!
d
being the Legendre-Jacobi-Kronecker symbol.
n
Then
lim(s − 1)ζK (s) = lim(s − 1)ζ(s)Ld (s) = lim Ld (s) = Ld (1),
s→1
s→1
s→1
since Ld (s) converges in the half-plane σ > 0 and lim(s − 1)ζ(s) = 1.
s→1
We obtain therefore,
!
∞
X
d 1
= κ · h.
Ld (1) =
n n
n=1
(64)
58
Applications to Algebraic Number Theory
54
For the determination of the left-hand side, we consider more generally, for
any character χ(, 1) of the ideal class group of K,
lim LK (s, χ) = LK (1, χ) =
s→1
Now,
X χ(a)
.
N(a)
a
59
κ
ζ(s, A) =
+ ρ(A) + · · · ,
s−1
and so
LK (s, χ) =
X
χ(A)ζ(s, A)
A
=
=
X
A
X
A
since, χ being , 1,
χ(A)
κ
+ ρ(A) + · · ·
s−1
χ(A)ρ(A) + terms involving higher powers of (s − 1),
P
A
χ(A) = 0. On taking the limit as s → 1, we have
LK (1, χ) =
X
χ(A)ρ(A).
(65)
A
Therefore, the problem of determination of LK (1, χ) has been reduced to that
of ρ(A).
We now study LK (s, χ) for a special class of characters, the so-called genus
characters (due to Gauss) to be defined below.
We call a discriminant, a prime discriminant, if it is divisible by only one
prime. In that case, d = ±p if d is odd, or if d is even, d = −4 or ±8.
Proposition 9. Every discriminant d can be written uniquely as a product of
prime discriminants.
Proof. It is known that if d is odd, then d = ±p1 . . . pk where p1 , . . . , pk are
mutually distinct odd primes. Let P1 = ±p1 according as p1 ≡ ±1( mod 4);
then P1 is a prime discriminant and d/P1 is again an odd discriminant. Using
induction on the number k of prime factors of d (odd!) it can be shown that
d/P1 = P2 , . . . , Pk where Pi are odd prime discriminants.
If d is even, then we know that
(a) d = ±8p2 , . . . , pk , if d/4 ≡ 2( mod 4),
Applications to Algebraic Number Theory
55
(b) d = ±4p2 , . . . , pk , if d/4 ≡ 3( mod 4),
p2 , . . . , pk being odd primes. Now we may write d = P1 d1 where P1 is chosen
to be −4, +8 or −8 such that d1 is an odd discriminant. From the above it is
clear that d = P1 P2 . . . Pk where P1 , . . . , Pk are all prime discriminants.
This decomposition is seen to be unique.
We may now introduce the genus characters.
Let d1 be the product of any of the factors P1 , . . . , Pk of d. Then d1 is again
a discriminant and d1 |d. Let d = d1 d2 ; d2 is also a discriminant. Indentifying
the decompositions d = d1 d2 and d = d2 d1 , we note that the number of such
decompositions (including the trivial one, d = 1 · d) is 2k−1 where k is the
number of different prime factors of d.
For any such decomposition d = d1 d2 of d and for any prime ideal p not
dividing d, we define
!
d1
χ(p) = χd1 (p) =
,
N(p)
!
d1
where
is the Legendre-Jacobi-Kronecker symbol. We shall show that
N(p)
!!
!
d2
d1
χd2 (p) =
=
.
N(p)
N(p)
60
In face, since p × d, p belongs to one of the following types. (a) pp′ = (p),
N(p) = p or (b) p = (p), N(p) = p2 . If pp′ = (p), N(p) = p then
!
!
!
d1
d2
d
=1=
p
N(p) N(p)
!
!
d2
d1
=
. If
which means that either both are +1 or both are −1. i.e.
N(p)
N(P)
p = (p), N(p) = p2 then dp = −1; but
!
!
!
d1
d2
d
= 2 = 1,
N(p) N(p)
p
!
!
d2
d1
=
.
so that we have again
N(p)
N(p) !
!
d2
d1
,
is zero, and the other nonWhen p|d, one of the symbols
N(p)
N(p)
zero, we take χ(p) to be the non-zero value. In any case, χ(p) = ±1. This
definition can then be extended to all ideals a of K as follows: If a = pk ql , . . .,
61
Applications to Algebraic Number Theory
56
we define χ(a) = (χ(p))k (χ(q))l , . . . , so that
χ(ab) = χ(a)χ(b)
for any two ideals a, b of K.
By a genus character, we mean a character χ defined as above, corresponding to any one of the 2k−1 different decompositions of d as product of two
mutually coprime discriminants.
We shall see later that the genus characters form an abelian group of order
2k−1 .
Now we shall obtain an interesting consequence of our definition of the
character χ, with regard to the associated L-series.
Consider the L-series, for s = σ + it, σ > 1
X
Y
LK (s, χ) =
χ(a)(N(a))−s =
(1 − χ(p)(N(p))−s )−1
a
=
YY
p
(1 − χ(p)(N(p))−s )−1 .
p p|(p)
The prime ideals p of K are distributed as follows:
!
d
= −1, N(p) = p2 ,
(a) p = (p),
p
!
d
(b) pp′ = (p),
= +1, N(p) = p,
p
!
d
= 0, N(p) = p.
(c) p2 = (p),
p
!
!
!
!
!
d1 d2
d1
d2
d
= −1 =
implies that one of
,
is +1 and the
In case (a),
p
p
p
p
p
other −1. So
Y
(1 − χ(p)(N(p))−s )−1 = (1 − p−2s )−1
p|(p)
= (1 − p−s )−1 (1 + p−s )−1
!−1
!−1
!
!
d1
d2
= 1−
1−
· p−s
· p−s
p
p
In case (b),
62
Applications to Algebraic Number Theory
57
!−1
!−1
!
!
d1
d1
−s
−s
1−
(1 − χ(p)(N(p)) ) = 1 −
·p
·p
p
p
p|(p)
!−1
!−1
!
!
d1
d2
−s
−s
= 1−
1−
·p
·p
,
p
p
!
!
!
!
!
d
d1 d2
d1
d2
since 1 =
=
implies that
=
. In case (c) p|d implies
p
p
p
p
p
that!p|d1 or p|d2 . We shall assume without loss of generality that p|d2 . Then
d2
= 0 and
p
!−1
!−1
!
!
Y
d2
d1
· p−s
· p−s .
× 1−
(1 − χ(p)(N(p))−s )−1 = 1 −
p
p
p|(p)
Y
−s −1
From all cases, we obtain
YY
LK (s, χ) =
(1 − χ(p)(N(p))−s )−1
p p|(p)
=
Y
p
!
!−1
!
!−1
d1
d2
−s
−s
1−
·p
1−
·p
p
p
= Ld1 (s)Ld2 (s).
We have therefore,
Theorem 4 (Kronecker). For a genus character χ of a K corresponding to
the decomposition d = d1 d2 , we have,
LK (s, χ) = Ld1 (s)Ld2 (s).
(66)
Let us consider the trivial decomposition d = 1 · d or d1 = 1 and d2 = d.
Then Ld1 (s) = ζ(s) and Ld2 (s) = Ld (s). In other words, (66) reduces to (63).
If d1 , 1, ζQ( √d1 ) (s) = ζ(s)Ld1 (s), from (63). But Ld1 (s) can then be continued
analytically into the whole plane and it is an entire function.
Two ideals a and b are said to be equivalent in the “narrow sense” if a =
b(γ) with N(γ) > 0. (For α ∈ K, N(α) denotes its norm over Q.)
This definition coincides with the usual definition of equivalence in the case
of an imaginary quadratic field since the norm of every element is positive. In
the case of a real quadratic field, if there exists a unit ǫ in the field with N(ǫ) =
−1, then both the equivalence concepts are the same, as can easily be seen.
Otherwise, if h0 denotes the number of classes under the narrow equivalence
and h, the number of classes under the usual equivalence, we have h0 = 2h.
63
Applications to Algebraic Number Theory
58
Proposition 10. For a genus character χ, χ(a) = χ(b), if a and b are equivalent
in the narrow sense.
Proof. We need only to prove that χ((α)) = 1 for α integral and N(α) > 0.
!
d1
(a) Let (α, d1 ) = (1). We have to show that
= 1. Since N(α) > 0,
! N((α))
d1
= 1.
this is the same as proving that
N(α)
!
d1
(1) Suppose d1 is odd. Then
= 1. If d is even, d = 4m (say). Then
4
√
√
α = x + y m with x, y rational integers. If not, 2α = x′ + y′ m
with x′ and y′ rational integers.! In any case, since (d!1 /4) = 1, it is
d1
d1
= 1 for two
sufficient to prove that
= 1. i.e. 2
N(2α)
a − mb2
rational integers a, b with a2 − mb2 > 0 and (a2 − mb2 , d1 ) = 1, in
both cases. Now d1 |d implies that d1 |m so that! from the periodicity
d1
of the Jacobi symbol, this is the same as 2 = 1 for (a, d1 ) = 1,
a
which is obvious.
(2) Suppose d1 is even. Then d1 = (even discriminant) × (odd discriminant) and we need consider only the even part, since we have
already disposed of the odd part in 1). We have then three possibilities d1 = −4, +8 and −8.
(i) d1 = −4. Then d = (−4)(−m) so that −m is again a discriminant and ≡ 1( mod 4).
Consider now
!
!
−4
d1
= 2
;
N(α)
x − my2
(x2 − my2 ) ≡ 1( mod 4) since both x, y cannot be even or odd,
for (α, d1 ) = (1). From the periodicity of the Jacobi symbol,
follows then that
!
!
−4
−4
=
= 1.
1
x2 − my2
(ii) d1 = +8. Here d = 8 × m/2; m/2 is then an odd discriminant
and ≡ 1( mod 4) or equivalently, m ≡ 2( mod 8).
Now,
!
!
8
d1
= 2
;
N(α)
x − my2
64
Applications to Algebraic Number Theory
59
(x2 − my2 ) ≡ ±1( mod 8) so that
!
!
!
!
8
8
8
1
=
=
1
or
=
=
= 1.
1
7
7
x2 − my2
2
2
If d1 = −8,! m ≡ −2(
! mod! 8) and x − my ≡ 1, 3( mod 8)
−8
1
d1
=
=
= 1. So, if (α, d1 ) = (1), χ((α)) =
and
N(α)
3
3
1. If (α, d1 ) , (1) and (α, d2 ) = (1), we can apply the same
arguments for d2 instead of d1 and prove χ((α)) = 1.
(b) If (α, d1 ) , (1) and (α, d2 ) , (1), we decompose (α) as follows: (α) =
p1 p2 . . . pl q where pi |d and (q, d) = (1). We choose in the narrow class of
p−1
1 , an integral ideal q1 with (d, q1 ) = (1). Then for the element α1 with
(α1 ) = p1 · q1 , N(α1 ) > 0, χ((α1 )) = 1, since (α1 , d1 ) or (α1 , d2 ) = (1).
Similarly for p2 , construct q2 with (d, q2 ) = 1 and for α2 with (α2 ) = p2 q2
and N(α2 ) > 0, χ((α2 )) = 1 and so on.
Finally, we obtain
(αα1 α2 . . .) = p21 p22 . . . p2l p
where (b, d) = (1). But pi |d imply that p2i = (pi ) with pi |d. Therefore (αα1 α2 . . .) =
(p1 p2 . . .)b so that b is a principal ideal = (ρ) (say). Now N(αα1 α2 . . .) =
p21 p22 . . . p21 N(ρ) and χ((αα1 . . . αl )) = χ((ρ)) = 1 since N(ρ) > 0 and (ρ, d) =
(1). But χ((α1 )) = 1, . . . χ((αl )) = 1 so that χ((α)) = 1. The proof is now 65
complete.
We have just proved that χ(a) depends only on the narrow class of a, say
A · χ is therefore a character of the ideal class group in the “narrow sense”.
Then,
X
X
X
LK (s, χ) =
χ(A) (N(a))−s =
χ(A)ζ(s, A)
(67)
A
a∈A
A
where A runs over all the ideal classes in the ‘narrow sense’.
Using a method of Hecke, we shall now give an alternative proof of the
fact that χ((α)) = 1 for principal ideals (α) with N(α) > 0 and (α, d) , (1).
Denote by χ0 that character of the ideal class group in the narrow sense,
such that χ0 (a) = χ(a) for all ideals a in whose prime factor decomposition,
only prime ideals p which do not divide d, occur. We need only to prove that
χ(a) = χ0 (a) for all ideals a.
Applications to Algebraic Number Theory
Let fχ0 (s) =
P
a
χ0 (a)(N(a))−s and fχ (s) =
60
P
a
χ(a)(N(a))−s . Then
fχ0 (s) Y (1 − χ(p)(N(p))−s )
=
,
fχ (s)
(1 − χ0 (p)(N(p))−s )
p|d
since the remaining factors cancel out. We shall show that the product on the
right is = 1, or fχ0 (s) = fχ (s).
P
We know that fχ (s) = Ld1 (s) = Ld2 (s) from (66) and fχ0 (s) = χ0 (A).
A
ζ(s, A) from (67). Both have functional equations of the same type, so that if we
denote by R(s), the quotient, the functional equation is simply R(s) = R(1 − s).
We shall now arrive at a contradiction, by supposing that χ , χ0 . For, then,
there exists a prime ideal p with p|d and χ(p) = ±1, χ0 (p) = ∓1.
log(∓1) + 2kπi
Choose −s =
(k, such that s , 0). This means that p−s =
log p
∓1. Consider the product
R(s) =
Y (1 − χ(p)(N(p))−s )
.
(1 − χ0 (p)(N(p))−s )
p|d
The above value of s is a zero of the denominator and it cannot be cancelled by
any factor in the numerator, for that would mean
log(∓1) + 2kπi log(±1) + 2lπi
=
,
log p
log q
or in other words, λ = log p/ log q is rational; i.e. qλ = p holds for p, q
primes and λ rational. This is not possible since p , q. By the same argument,
the zeros and poles of R(s) cannot cancel with those on the other side of the
equation R(s) = R(1 − s). But this is a contradiction. In other words, χ(a) =
χ0 (a) for all ideals a of K.
Tow ideals a and b are in the same genus, if χ(a) = χ(b) for all genus
characters χ defined as above, associated with the decompositions of d.
The ideals a for which χ(a) = 1 for all genus characters χ, constitute the
principal genus. Then the narrow classes A for which χ(A) = 1 are the classes
in the principal genus. If H denotes the group of narrow classes and G, the
subgroup of classes lying in the principal genus, the quotient group G = H/G
gives the group of different genera. We claim that the order of G is 2k−1 , k
being the number of different prime factors of d. For, suppose we have proved
that the genus characters form a group of order 2k−1 . From the definition of a
genus and from the theory of character groups of abelian groups, we know that
66
Applications to Algebraic Number Theory
61
this group is the character group of G. By means of the isomorphism between
G and its character group, it would then follow that the order of G is also 2k−1 .
If now remains for us to prove
Proposition 11. The genus characters form an abelian group of order 2k−1 ,
where k is the number of distinct prime factors of d.
Proof. Now, χ(a) = ±1 implies that χ2 = 1 for all genus characters χ. So,
the character χ−1 exists and = χ. We shall prove that the genus characters and
closed under multiplication and that they are all different for different decompositions of d.
Let d = d1 d2 = d1∗ d2∗ . Set d1 = qu1 and d1∗ = qu∗1 so that q = (d1 , d1∗ ) and
(u1 , u∗1 ) = 1.
If d3 = u1 u∗1 then d1 d1∗ = q2 d3 . Also for the characters χd1 and χd1∗ associated with the two decompositions d1 d2 and d1∗ d2∗ of d, we have χd1 χd1∗ = χd3 and
d3 |d, since both d1 |d and u∗1 |d and they are coprime. Hence χd3 is again a genus
character. Now, if d1 , d1∗ , we have to show that χd1 , χd1∗ or χd1 χd1∗ , χ1 (the
identity character); or in other words, we need only to prove that for a proper
decomposition d = d3 d3∗ , the associated character χ is not equal to χ1 .
We have, from (66), for σ > 1,
X
LK (s, χ) = Ld3 (s)Ld3∗ (s) =
χ(A)ζ(s, A).
A
Comparing the residue at s = 1 in the two forms for LK (s, χ), we have
P
χ(A) =
A
0, which implies that χ , χ1 .
Thus the genus characters are different for different decompositions of d
and they form a group. Since there are 2k−1 different decompositions of d, this
group of genus characters is of order 2k−1 .
A genus character is a narrow class character of order 2. Conversely, we
can show that every narrow class character of order 2 is a genus character. For
this, we need the notion of an ambiguous (narrow) ideal class.
A narrow ideal class A satisfying A = A′ (the conjugate class of A in K) or
equivalently A2 = E (the principal narrow class) is called an ambiguous ideal
class. The ambiguous classes clearly form a group.
√
Proposition 12. The group of ambiguous ideal classes in Q( d) is of order
2k−1 , k being the number of distinct prime factors of d.
Proof. We shall pick out one ambiguous ideal b (i.e. such that b = b′ , the
conjugate ideal) from each ambiguous ideal class A and show that there are
2k−1 such inequivalent ideals.
67
Applications to Algebraic Number Theory
62
Let a be any ideal in the ambiguous ideal class A. Since a ∼ a′ , we may
take aa′ −1 = (λ) with N(λ) = 1 and λ totally positive (in symbols; λ > 0). (For
d < 0, the condition “totally positive” implies no restriction). Let ρ = 1 + λ.
Then ρ , 0 since λ , −1. Further λ = (1 + λ)/(1 + λ′ ) = ρ/ρ′ . If we define
b = a/(ρ), the b = b′ , for b/b′ = a/a′ (ρ′ )/(ρ) = (1). Further b ∈ A, since ρ > 0
(i.e. ρ > 0, ρ′ > 0). We may take b to be integral without loss of generality.
We call b primitive, if the greatest rational integer r dividing b is 1. For any
ambiguous integral ideal b we have b = rb1 with r, the greatest integer dividing
b and b1 primitive. If b ∈ A, b1 ∈ A.
We shall now prove that any primitive integral ambiguous ideal b is of the
form pλ11 , . . . , pλk k with λi = 0 or 1 and pi |ϑ, (pi , p j ), where ϑ is the different of
√
K = Q( d) over Q. Let b = qµ11 , . . . , qµs s . Now b = b′ implies that qµ11 , . . . , qµs a =
µ
q′ 11 , . . . , q′ µs s . We have then qµ11 = q′ µt t or µ1 = µt and q1 = q′t . We assert then
t = 1, for if not, q1 = q′t implies that q′1 = qt and the factor qµ11 qµt t = (q1 q′1 )µ1 is a
rational ideal , (1). This is a contradiction to the hypothesis that b is primitive.
Hence t = 1 and q1 = q′1 . The same argument applies to all prime ideals qi .
Since q2i = (pi ) with prime numbers pi , the exponents µi are either 0 or 1.
for otherwise they bring in rational ideals which are excluded by the primitive
nature of b. Therefore, we have b = pλ11 , . . . , pλk k with pi |ϑ and λi = 0 or 1.
Now, we shall show that there are exactly 2k−1 inequivalent (in the narrow
sense) such ideals. For the same, it is enough to prove that there is only one
non-trivial relation of the form pl11 , . . . , plkk ∼ (1) in the narrow sense, for, then,
it would imply that among the 2k such ideals, there are 2k−1 inequivelent ones,
which is what we require.
We shall first prove uniqueness, namely that if there is one non-trivial relation, then it is uniquely determined. Later, we show the existence of a nontrivial relation.
P
(a) Let pl11 , . . . , plkk = (ρ) with ρ > 0 and li > 0 or (ρ) , (1). Then the set
i
(l1 , . . . , lk ) is uniquely determined.
For, (ρ) = (ρ′ ) implies that ρ = η · ρ′ with a totally positive unit η.
The group of totally positive units in K being cyclic, denote by ǫ, the
generator of this group. (Note that, for d < 0, the condition “totally
positive” imposes no restrictions). Since (ρ) = (ρǫ n ), when ρ is replaces
by ρǫ n , η goes over to ηǫ 2n . Choosing n suitably, we may assume without
loss of generality η = 1 or ǫ. We shall see that (ρ) , (1) implies that
η , 1. For, if η = 1, then ρ = ρ′ = a natural number. Now (N(ρ)) =
2lk
lk
l1
1
(ρ2 ) = p2l
1 , . . . , pk = (p1 , . . . , pk ), if N(pi ) = pi , pi , p j and ρ being a
68
Applications to Algebraic Number Theory
63
natural number, this is possible only if all li = 0, in which case (ρ) = (1).
But that is a contradiction to the hypothesis (ρ) , (1).
We are therefore left with the case η = ǫ. √
Then set µ = 1 − η = 1 − ǫ , 0.
We have µ = −ηµ′ . Denoting by δ = d, (ρ/µδ)′ = ρ/µδ = r = a
rational number , 0. Hence (ρ) = (µδ)(r) and the decomposition
!
!
1 l1
1
(ρ) =
p , . . . , plkk
(µδ) =
r
r 1
is uniquely determined or in other words, the set (l1 , . . . , lk ) is uniquely
fixed.
(b) We shall now prove the existence of a non-trivial relation pl11 , . . . , plkk ∼
(1)(li = 0 or 1) in the narrow sense.
Consider µ = 1 − ǫ. Then µδ is in general not primitive. Now choose
a rational number r suitably so that rµδ is primitive and denote it by ρ.
Then rµδ = ρ = ǫρ′ . Now if d < 0, N(µδ) > 0 clearly and if d > 0,
N(µδ) = dǫµ′ 2 > 0, again. Hence N(ρ) > 0. We may assume that ρ > 0
and since ǫ is not a square, ρ is not a unit, so that (ρ) , (1). Since (ρ) is
primitive, (ρ) , rational ideal. Now (ρ) = (ρ′ ), ρ > 0 and (ρ) is primitive
so that pl11 , . . . , plkk = (ρ) ∼ (1) in the narrow sense. Further this relation
is non-trivial as we have just shown. Proposition 12 is thus completely
proved.
Now, the group of narrow class characters χ with χ2 = 1 is again of
order 2k−1 , since it is isomorphic to the group of ambiguous (narrow)
ideal classes in K. But the genus characters from a subgroup (of order
2k−1 ) of the group of narrow class characters χ with χ2 = 1. Hence every
narrow class character of order 2 is a genus character. Or, equivalently
every real narrow class character is a genus character.
For any genus character χ, χ(i2 ) = 1 for every ideal i. In other words,
i2 is in the principal genus. Or, for any two ideals a and b with a ∼ bi2
(in the narrow sense), we have χ(a) = χ(b) for every genus character χ.
That the converse is also true is shown by
Theorem 5 (Gauss). If two ideals a and b are in the same genus, there exists
an ideal i such that a ∼ bi2 , in the narrow sense. In particular, if a is in the
principal genus, a ∼ i2 for an ideal i.
Proof. Two narrow classes A and B are by definition, in the same genus if and
only if χ(A) = χ(B) for all genus characters χ or by the foregoing, for all narrow
class characters χ with χ2 = 1. But from the duality theory of subgroups of
69
Applications to Algebraic Number Theory
64
abelian groups and their character groups, we deduce immediately that AB−1 =
C 2 for a narrow ideal class C. This is precisely the assertion above.
Remarks. (1) The notion of genus in the theory of binary quadratic forms
is the same as above, if one carries over the definition by means of the
correspondence between ideals and quadratic forms.
(2) Hilbert generalized the notion of genus to arbitrary algebraic number
fields and used it for higher reciprocity laws and class field theory.
We now come back to formulas (66) and (67) (d < 0). If d = d1 d2
X
Ld1 (s)Ld2 (s) =
χ(A)ζ(s, A),
A
√
the summation running over all (narrow) ideal classes A in K = Q( d).
Consider any ideal b ∈ A−1 . For a ∈ A, ab = (γ) with N(γ) > 0 we have
then
(N(b)) s X
ζ(s, A) =
(N(γ))−s ,
w b|γ,0
√
where w denotes the number of units in K = Q( d).
Let [α, β] be an integral basis of b. Then we may write b = [α, β] =
(α)[1, β/α] so that β/α = z = x + iy with y > 0, i.e. we may suppose that
b = [1, z]. Then, for γ ∈ b, if γ = m + nz with m, n integers , (0, 0),
N(γ) = |m + nz|2 . We obtain
ζ(s, A) =
N([1, z]) s X′
|m + nz|−2s
w
m,n
√
Now, abs. 11 zz′ = |z′ − z| = 2y = N(b) |d|, so that
1 2y
ζ(s, A) =
√
w
|d|
=
!s X
m,n
1 2
2
√
√
w |d|
|d|
′
|m + nz|−2s
! s−1
ys
′
X
m,n
|m + nz|−2s .
From Kronecker’s first limit formula, we obtain the following expansion:
!
X′
1
√
−2s
s
2
|m + nz| = π
y
+ 2C − 2 log 2 − 2 log( y|η(z)| ) + · · · .
s−1
m,n
70
Applications to Algebraic Number Theory
65
Further
71
2
√
|d|
! s−1
= e(s−1) log 2/
√
|d|
!
2
= 1 + (s − 1) log √ + · · · .
|d|
On multiplication, we obtain
ζ(s, A) =
We now set
p
1
2π
+ 2C − log |d| − 2 log( N([1, z])|η(z)|2 )+
√
s
−
1
w |d|
+ terms with higher powers of (s − 1) .
F(A) =
p
N([1, z])|η(z)|2 .
(∗)
Clearly, F(A) depends only upon the class A. In case d < −4, w = 2, so that
−2π X
χ(A) log F(A)+
Ld1 (s)Ld2 (s) = √
|d| A
+ terms with higher powers of (s − 1).
Taking s = 1, we have
−2π X
Ld1 (1)Ld2 (1) = √
χ(A) log F(A).
|d| A
Let us assume without loss of generality, d1 > 0, d2 < 0. Then, by Dirichlet’s class number formula for a quadratic field, we have
Ld1 (1) =
2h1 log ǫ
√
d1
√
whre ǫ is the fundamental unit and h1 the class-number of Q( d1 ) and
Ld2 (1) =
2πh2
√ ,
w |d2 |
√
h2 denoting the class-number and w, the number of roots of unity in Q( d2 ).
On taking the product of these two, we obtain
X
2h1 h2
log ǫ = −
χ(A) log F(A).
w
A
Summing up, we have the following. Let d = d1 d2 < −4 be the discriminant of an imaginary quadratic field over Q and let d1 > 0, d2 < 0 be again discriminants of quadratic fields over Q with class-numbers
h1 , h2 , respectively.
√
Let w be the number of roots of unity in Q( d2 ). Then one has
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Applications to Algebraic Number Theory
66
√
Theorem 6. For the fundamental unit ǫ of Q( d1 ), we have the formula
Y
ǫ 2h1 h2 /w =
(F(A))−χ(A) ,
(68)
A
√
where A runs over all the ideal classes in Q( d) and F(A) has the meaning
given in (∗) above.
If w = 2, the exponent of ǫ in (68) is a positive integer.
Now ǫ satisfies Pell’s diophantine equation x2 − d1 y2 = ±1. And F(A)
involves the values of |η(z)|. So we have a solution of Pell’s equation by means
of “elliptic functions”. This was found by Kronecker in 1863.
An Example. We shall take d = −20, d1 = 5, d2 = −4√so that d = d1 d2 . We
know that h1 = h2 = 1, w = 4. The class number of Q( −20) is 2 and for the
two ideal classes, say A1 , A2 , we can choose as representatives the ideals
√ 

√
 1 + −5 
 .
b1 = [1, −5] = (1) and b2 = 1,
2
Now N(b1 ) = 1 and N(b2 ) = 1/2. From the definition of F(A), we have
√
2
F(A1 ) = |η( −5)|
and

√ 2
1  1 + −5 
 .
F(A2 ) = √ η 
2
2
If χ(, 1) be the genus character associated !with the above decomposition of d,
5
= −1. Formula (68) now takes the
then necessarily χ(A1 ) = 1 and χ(A2 ) =
2
form

√  4
 1 + −5 


η 
2
1
ǫ=
.
(69)
√
2 |η( −5)|4
From the product expansion of the η-function, we obtain
√
√
η( −5) = e−π 5/12 |(1 − q2 )(1 − q4 )(1 − q6 ) . . . |,

√ 
√
 1 + −5 
−π 5
, the following expansion:
and for η 
where q = e
2

√ 
√
 1 + −5 
 = e−π 5/24 |(1 + q)(1 − q2 )(1 + q3 )(1 − q4 ) . . . |.
η 
2
73
Applications to Algebraic Number Theory
67
On substituting these expansions in (69), we have
ǫ=
1 π √5/6
(1 + q)4 (1 + q3 )4 (1 + q5 )4 . . . .
e
2
√
Now, eπ 5/3 > 10 so that q < 10−3 or q3 < 10−9 . Hence the product on the
right converges rapidly and if we replace the infinite product by 1, we have
ǫ=
1 π √5/6
e
+ O(10−2 )−
2
i.e.
√
1 + 5 1 π √5/6
∼ e
(upto an error of the order of 10−2 ).
ǫ=
2
2
It is to be noted that in the expression for ǫ as an infinite product, if one
cuts off at any finite stage, the resulting number on the right is always transcendental, but the limiting value ǫ on the left is algebraic.
We consider now the trivial decomposition
X
ζd (s) = ζ(s)Ld (s) =
ζ(s, A).
A
On substituting the expansion of ζ(s, A) on the right side in powers of (s − 1),
we have



2X
2πh  1
+ 2C − log |d| −
ζd (s) = √ 
log F(A) + · · ·  .
(70)
h A
w |d| s − 1
!
∞ d
P
n−s =
On the other hand, ζ(s) = 1/(s − 1) + C + · · · and Ld (s) =
n=1 n
L(1) + (s − 1)L′ (1) + · · · . From (70), we have now the equation,



2πh  1
2X
+ 2C − log |d| −
log F(A) + · · · 
√ 
h A
w |d| s − 1
!
1
=
+ C + · · · (L(1) + (s − 1)L′ (1) + · · · ).
s−1
On comparing coefficients of 1/(s − 1) and constant terms on both sides, we
obtain

2πh



L(1) = √ ,



w |d|


 
(71)

X



2πh 
2
 


L′ (1) = √ C − log |d| −
log F(A) ,


h A
w |d|
74
Applications to Algebraic Number Theory
68
or we have an explicit expression for
L′ (1)
2X
= C − log |d| −
log F(A).
L(1)
h A
(71)′
We shall obtain now for the left side, an infinite series expansion.
Let us consider the infinite product
!−1
!
Y
d −s
.
p
1−
L(s) =
p
p
this product converges absolutely for σ > 1, and on taking logarithmic derivatives, we obtain
!−1 !
!
X
d −s
d −s
L′ (s)
1−
p
p log p
=−
L(s)
p
p
p
!
d
X p log p
!.
=−
(72)
d
p ps −
p
On passing to the limit as s → 1, if the series on the right converges at
s = 1, then by an analogue of Abel’s theorem, it is equal to L′ (1)/L(1). But it
is rather difficult to prove. One may proceed! as follows:
d
= 1} and π− (X) = {p : p prime
Let π+ (X) = {p : p prime ≤ X and
p
!
d
≤ X and
= −1}. Then π+ (X) − π− (X) tends to ∞ less rapidly than π(X)
p
as X → ∞. If one could show that this function has the estimate 0(X/(log X)′ )
where r > 2, then the convergence of the series on the right side of (72) at
s = 1, can be proved. For this estimate, one requires a generalization of the
proof of prime number theorem for arithmetical series.
We shall now compute L′ (1)/L(1) for some values of d.
√
√
Examples. d = −4. The quadratic field Q( d) = Q( −1) with discriminant
−4 has class number 1. Also,
!
d
= 1 if n ≡ 1 (mod 4) and = −1 if n ≡ 3 (mod 4).
n
Therefore,
L(s) = 1−s − 3−s + 5−s − · · ·
75
Applications to Algebraic Number Theory
69
and
L(1) = 1−1 − 3−1 + 5−1 − · · ·
the so-called Leibnitz series. From Dirichlet’s class number formula, we have
L(1) =
2π
π
= = 1−1 − 3−1 + 5−1 − · · · .
4.2 4
More interesting is the series for L′ (1)/L(1):
L′ (1) log 3 log 5 log 7 log 11 log 13
=
−
+
+
−
− ···
L(1)
4
4
8
12
12
= (C − log 4 − 4 log η(i)).
Now
η(i) = e−π/12
∞
Y
n=1
We have 6−2π < 1/400 and hence
4
∞
Q
n=1
∞
P
(1 − e−2πn ).
e−nπ is rapidly convergent. In other words
n=1
(1 − e−2nπ ) is absolutely convergent and upto an error of 10−2 , it can be
replaced by 1. We obtain consequently
π
L′ (1) ∼ C − log 4 +
.
L(1)
3
2 Class
√ number of the absolute class field of
Q( d)(d < 0).
We shall now apply the method outlined
√ in § 57 to determine the class number
of the absolute class field of K0 = Q( d), with d < 0. But, for the time being,
however, let K0 be an arbitrary algebraic number field. The absolute class field
K/K0 is, by definition, the largest field K containing K0 , which is both abelian
and unramified over K0 (i.e. with relative discriminant over K0 equal to (1)).
The absolute class field, first defined by Hilbert, is also known as the
Hilbert class field; its existence and uniqueness were proved by Furtwängler. It
can be shown that there are only finitely many abelian unramified extensions of
K0 and all these are contained in one abelian extension and this is the maximal
one. It has further the property that the Galois group G(K/K0 ) is isomorphic to
the narrow ideal class group of K0 .
76
Applications to Algebraic Number Theory
70
Consider now any abelian unramified extension K1 of K0 . Then G(K1 /K0 )
is isomorphic to a factor group of G(K/K0 ), i.e. to a factor group of the narrow
ideal class group of K0 . The character group of G(K1 /K0 ) is a subgroup of the
group of h0 ideal class characters in the narrow sense.
K
K1
K0
Q
We have then for the zeta function,
Y
ζK1 (s) =
L(s, χ),
77
χ
the product on the right running over all characters in this subgroup.
We shall now derive an expression for the quotient
H/h of the class num√
bers H of K and h of K0 in the case K0 = Q( d) with d < 0, by using the
above product formula for K1 = K and comparing the residue at s = 1 on both
sides.
We have indeed
Y′
L(s, χ) · ζd (s),
ζK1 (s) =
Q′
χ
the product
running over all characters χ , 1.
On comparing the residues at s = 1 on both sides, we obtain
2r1 (2π)r2 R
2πh Y
H= √
L(1, χ).
√
W |D|
w |d|
′
(73)
Here r1 and r2 denote the number of real and distinct complex conjugates of
the field K1 . The regulator of K1 is denoted by R and is defined as follows: By
a theorem of Dirichlet, every unit ǫ of K1 is of the form ǫ = ǫ0g0 ǫ1g1 . . . ǫrgr with
rational integers gi , ǫ0 being a root of unity. Then ǫ1 , . . . , ǫr (with r = r1 +r2 −1)
are called fundamental units. Define integers ek as follows. If K1(1) , . . . , K (r1 )1
are the real conjugates of K1 and K1(r1 +1) , . . . , K1(r1 +r2 ) the complex ones, then
we define



1 if k ≤ r1
ek = 

2 if r1 < K ≤ r.
Applications to Algebraic Number Theory
71
Then R = det(ek log |el(k) |)(l, k = 1 to r); here ǫl(k) denoting the conjugates of ǫl
in the usual order.
The regulator R is then a real number , 0 and without loss of generality,
one may assume R to be positive. It can then be shown that R is independent
of the choice of the fundamental units. In the case of the rational number field
and an imaginary quadratic field, R is by definition, equal to 1. Further in (73),
D denotes the absolute discriminant of K1 and W, the number of roots of unity
in K1 .
We shall now take K1 = K, i.e. the absolute class field. Then K is totally
imaginary of degree 2h, i.e. r1 = 0, r2 = h; since K is unramified over K0 ,
D = dh . We shall further assume that d < −4. Then (73) may be rewritten as



(2π)h RH
πh Y′ 2π  X
= √
χ(A) log F(A)
√ −
h/2
W|d|
|d| χ
|d|
A
or


Y′  X

2RH


χ(A) log F(A) .
=h
−
W
χ
A
(74)
With every element Ak of a finite group {A1 , . . . , Ah }, we associate an indeterminate uAk (k = 1, . . . , h). Then the determinant |uA−1
| is called the group
k Al
determinant and was first introduced by Dedekind and extensively used by
Frobenius. In the case of an abelian group we have then a decomposition of
this group determinant, as follows:

 

Y X

 X  Y X


u −1 =
χ(A)uA  .
 χ(A)uA  =  uA  
Ak Al
χ
A
A
χ,1
A
Let us suppose Ah = E (the identity); we can show by an elementary transformation that
X  u −1 =  uA  u −1 − u −1 (k, l = 1 to h − 1).
Ak Al
A
A
A
 k t

k
A
Then, from the above, we deduce that

Y X


u −1 − u −1 =
 χ(A)uA  .
Ak
Ak At
χ,1
Taking uA = − log F(A), we obtain
A
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Applications to Algebraic Number Theory
72




Y  X

F(A−1


k ) 
−

χ(A) log F(A) = det log


−1
F(Ak Al )
A
χ,1


p


2
N(bAk )|η(zAk )|


 = ∆(say).
= det log q

N(b −1 )|η(Z −1 )|2 
Ak At
Ak At
From 74, we gather that
79
2H
∆
= .
(75)
W ·h R
In other words, ∆/R is a rational number. We shall now discuss the (h − 1)rowed determinant ∆.
Now, bhA−1 is a principal ideal = (βl ) (say) with βl ∈ K0 . Then N(bhA−1 = |βl |2
l
)h/2 = |βl |. Define
so that N(bA−1
l
ρl =
l
η24h (zE )
(l = 1 to h − 1).
24h z −1
β12
Al
l η
We shall now prove that ρl are all units in the absolute class field K. For
the same, we first see that ρl depends only upon the class A−1
l . If b is replaced
by b(λ) with λ ∈ K0 , then
ρl →
η24h (1, zE )
= ρl
(β1 λ12h )λ−12h η24h 1, zA−1
l
since, in the “homogeneous” notation, η24 (λω1 , λω2 ) = λ−12 η(ω1 , ω2 ).
Now, we pick out a prime ideal p in the class A−1
l with the property that
′
pp = (p), p , p′ ; such prime ideals always exist in each class (, E) as a
consequence of Dirichlet’s theorem. We shall show that the ρl defined with
respect to p are units in the absolute class field K.
Let (p) = B1 , . . . , Br be the decomposition of p in K. If we denote degK0 Bi =
ν, then r · v = h. The ideal pv is principal and equal to (β) (say). Let [ω1 , ω2 ]
be an integral base of K0 and [ω∗1 , ω∗2 ] an integral base of p. Then we have
(ω∗1 ω∗2 ) = (ω1 ω2 )Pv where Pv is a 2-rowed square matrix of rational integers
and |Pp | = p.
Denoting by
η24 ((ω1 , ω2 )Pp )
ϕPp (ω) = p12 24
η ((ω1 , ω2 ))
(with ω = ω2 /ω1 ), we know from the theory of complex multiplication that
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Applications to Algebraic Number Theory
73
ϕPp (ω) ∈ K and the principal ideal
ϕPp (ω) = (p′ 12 )
(meaning the extended ideal in K). On taking the vth power on both sides,
12
ϕPp (ω)v = (p′ 12v ) = (p′ v )12 = (β )
or in other words,
v
12
ϕPp (ω) = β ǫ with ǫ, a unit in K.
Writing out explicitly, we have, finally, since |β|2 = Npv = pv ,
β12 η24v ((ω1 ω2 )Pp )
= ǫ, a unit in K.
η24v ((ω1 ω2 ))
The same also holds for a suitable unit ǫ, also with h instead of v since v|h.
(See references (2) Deuring, specially pp. 32-33, (4) Fricke, (5) Fueter).
If σ : α(h) → α(k) (k = 1 to h) denotes the automorphism of K/K0 corresponding to the class Ak under the isomorphism between G(K/K0 ) and the
ideal class group of K0 , it can be shown that

12h
p


2
N(bAk )|η(zAk )|


(k)
 (k = 1 to h − 1).
|ρl | =  q
 N(b −1 ) · |η(z −1 )|2 
Ak Al
Ak Al
We may then rewrite (75) as
12h−1 hh−2 2h H = W
det(log |ρl(k) |)
det(log |ǫl(k) |)
(k, l = 1 to h − 1),
(76)
where ǫ1 , . . . , ǫh−1 are a system of fundamental units and ǫl(k) , ρl(k) denote the
conjugates of ǫl and ρl (l = 1 to h − 1) respectively, the conjugates being chosen
in the manner indicated above. Since the number on the left is strictly positive,
the (h−1) units ρ1 , . . . , ρh−1 are independent, i.e. they have no relation between
them. Hence the group generated by these units is of finite index in the whole
unit group and the index is precisely given by the integer on the left side of
(76).
We have thus proved
81
Applications to Algebraic Number Theory
74
Theorem√7. Let K be the absolute class field of an imaginary quadratic field
K0 = Q( d), d < −4, H, h the class numbers of K and K0 respectively, R the
regulator of K and W the number of roots of unity in K. Then for H we have
the formula
H W ∆
=
·
h
2 R
where ∆ = |(log |ρl(k) |1/12h )|, l, k = 1, 2, . . . , h − 1 and ρl(k) are conjugates of h − 1
independent units ρ1 , . . . , ρh−1 in K, as found above.
√
Example. We shall take d = −5, i.e. K0 = Q( −5).
The class number h of K0 is 2.
√ The absolute
√ class
√ field K is then a biquadratic field and is given by K0 ( −1) = Q( −5, −1). Further,
√ √
Q( 5, −1) = K
2
√
Q( −5) = K0
2
Q
√
W = 4 and the fundamental unit is ǫ = (1 + 5)/2. We take A1 , A2 (= E)
to be the two ideal classes of K0 and the unit ǫ1 √
= ǫ −1 . The bases for the
√ two
ideal representatives are given by bA1 = [1, (1 + −5)/2] and bA2 = [1, −5].
(Refer to the example on page 71. Then from the above, we have
12



 2|η( √−5)|4 

|ρ1 | =  √ 4 
 η 1+ −5 
2
and from (69), we have then |ρ1 | = |ǫ1 |12 . From (76), we gather then that
12.22 · H = 4 ·
log(|ρ1 |)
= 4.12 or H = 1.
log(|ǫ1 |)
We shall consider, later, class numbers of more general relative abelian
extensions (which are not necessarily unramified) of imaginary quadratic fields
by using Kronecker’s second limit formula.
82
Applications to Algebraic Number Theory
75
3 The Kronecker Limit Formula for real quadratic
fields and its applications
In the last section, we applied Kronecker’s first limit formula to determine the
constant term in the expansion at s = 1 of the zeta-function ζ(s, A) associated
with an ideal class A of an imaginary quadratic field over Q. In what follows,
√
we shall consider first, a similar problem for a real quadratic field K0 = Q( d),
d > 0. This problem which is more difficult, was solved by Hecke in his paper,
“Über die Kroneckersche Grenzformel für reelle quadratische Körper und die
Klassenzahl relativ-abelscher Körper”. We shall, however, follow a method
slightly different from Hecke’s.
We start from the series
f (z, s) = y s
∞
X
′
m,n=−∞
−|m + nz|−2s ,
with z = x + iy, y > 0 and s = σ + it, σ > 1. It is easy to verify that
f (z, s) is a ‘non-analytic modular function’, i.e. for a modular transformation
z → z∗ = (αz + β)/(γz + δ), we have f (z∗ , s) = f (z, s).
We now make a remark which will not be used later, but which is interesting
in itself, namely, f (z, s) satisfies the partial differential equation
y2 ∆ f = s(s − 1) f
(77)
where ∆ = (∂2 /∂x2 ) + (∂2 /∂y2 ) is the Laplace operator.
For proving this, let us first observe that if F(z, w) is a complex-valued
function, twice differentiable in z and w and if (z, w) → (z∗ , w∗ ) where z∗ (αz +
β)(γz + δ)−1 , w∗ = (αw + β)(γw + δ)−1 with α, β, γ, δ real and αδ − βγ = 0, then
it can be shown directly by computation that
(z − w)2
∂2 F(z∗ , w∗ )
∂2 (F(z, w))
.
= (z∗ − w∗ )2
∂z∂w
∂z∗ ∂w∗
Setting w = z and αδ−βγ > 0, we see that y2 ∆ is an operator invariant under the
transformation z → z∗ . Now consider the function y s . It satisfies the equation
y2 ∆(y s ) = s(s − 1)y s .
If we use the invariance property of y2 ∆, then we see that
y2 ∆(y∗s ) = s(s − 1)y∗s .
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Applications to Algebraic Number Theory
76
Since y∗ = (αδ − βγ)y · |γz + δ|−2 , we have
y2 ∆(y s |γz + δ|−2s ) = s(s − 1)y s |γz + δ|−2s
where γ, δ are real numbers not both zero.
It is now an immediate consequence that f (z, s) satisfies the equation (77).
P
In fact, it also follows that the series y s ′ |m + nz|−2s e2πi(mu+nv) satisfies the
(m,n)
differential equation (77).
We shall now derive some properties of a function F which satisfies the
equation (77).
Suppose the function F itself can be written in the form,
F = (F−1 )/(s − 1) + F0 + F1 (s − 1) + · · · ;
then on setting s(s − 1)F = (s − 1)2 F− + (s − 1)F in (77) = and by comparing
coefficients, we obtain the recurrence relation
y2 ∆Fn = Fn−1 + Fn−2 , n = 0, 1, 2, . . . ,
where F−2 = 0.
Now, consider the function f (z, s). From Kronecker’s first limit formula,
F−1 = π and from the above recurrence relation, we obtain y2 ∆F0 = π. If we
write F0 = −π log y + G, then y2 ∆G = 0. In other words, G = F0 + π log y is a
potential function. This is also explained by the fact that
G = 2π(C − log 2 − log |η(z)|2 ).
P
For the function F(z, s) = y s ′ m,n |m + nz|−2s e2πi(mu+nv) , from Kronecker’s
second limit formula, F−1 = 0 so that y2 ∆F0 = 0, i.e. F0 is a potential function,
which can also be seen directly from the expression for F0 .
These remarks stand in connection with the work of Maass and Selberg on
Harmonic analysis.
Now, let αγ βδ be the matrix of a hyperbolic substitution having two real
fixed points ω, ω′ (ω , ω′ ), both being finite. We shall assume without loss of
generality, that ω′ < ω. Set u = (z − ω)(z − ω′ )−1 . Then u∗ = (z∗ − ω)(z∗ − ω′ )−1 .
The transformation z → z∗ corresponds to the transformation u → u∗ = λu with
λ, a positive real constant , 1. We can also assume without loss of generality
that λ > 1 (otherwise, we may take the inverse substitution). If we introduce a
new variable v,by defining u = λv , then the transformation u → u∗ = λu goes
over to v → v∗ = v + 1.
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Applications to Algebraic Number Theory
77
Consider a non-analytic function f of z, invariant under the substitution
z → z∗ = (αz + β)(γz + δ)−1 . Then f considered as a function of v, has period
1. If v = v1 + iv2 , then f (z) = g(v1 , v2 ) has period 1 in v1 .
log |u|
Now |u| = λv1 or v1 =
so that e2πiv1 = |u|2πi/ log λ . If g(v1 , v2 ) possesses
log λ
a valid Fourier expansion with respect to v1 , it is of the form
f (z) = g(v1 , v2 ) =
=
∞
X
n=−∞
∞
X
n=−∞
cn (v2 )e2πinv1
c∗n
u
u
|u|2πin/ log λ
(78)
log u/u
. In general, the Fourier coefficients c∗n in (78), are not
2i log λ
constants but if u has a constant argument, i.e. if u/u is a constant, then c∗n are
constants.
We shall compute the Fourier coefficients √
c∗n of f (z, s) in the case when ω,
′
ω come from a real quadratic field K0 = Q( d) with discriminant d > 0. It
is interesting to see that upto certain factors, the Fourier coefficients are just
Hecke’s “zeta-functions with Grössencharacters” ssociated with K0 .
Suppose ǫ is a non-trivual (, ±1) unit in K0 . Let [ω, 1] be an integral basis
of an ideal b in K0 . Without loss of generality, we may suppose that ω > ω′
(otherwise −ω has this property!).
Now (ǫ)b = b so that ǫω = αω + β, ǫ = γω + δ with rational integers α, β,
γ, δ. We may also write
!
!
!
!
α β ω ω′
ω ω′ ǫ 0
=
1 1 0 ǫ′
γ δ 1 1
since v2 =
It follows then that αδ − βγ = ǫǫ ′ = ±1. In the case αδ − βγ = 1, define u
ωu + ω′
αz + β
ωu∗ + ω′
such that z =
or if z∗ =
, then z∗ =
; in other words,
u+1
γz + δ
u∗ + 1
′
z−ω
z − ω′
u =
and u∗ = ǫ 2 u. If αδ − βγ = N(ǫ) = −1, define u =
and
ω−z
ω−z
αz + β ∗
, u = −ǫ 2 u. Since z → z∗ is a hyperbolic transformation, the
if z∗ =
γz + δ
points ω′ , z, z∗ , ω lie on the same segment of a circle in the case N(ǫ) = 1. We
may assume without loss of generality, that z∗ lies to the right of z, or otherwise,
we can take the reciprocal substitution. We have then ǫ 2 > 1, since ǫ > ǫ ′ , as
a consequence of our assumption that z∗ lies to the right of z · ǫ itself might be
positive or negative. If ǫ is negative, we take instead of α, β, γ, δ, the integers
85
Applications to Algebraic Number Theory
78
−α, −β, −γ, −δ so that we secure finally that ǫ > 1. In the second case, when
N(ǫ) = −1, if z lies on the small segment ω′ λω (see figure), z∗ would lie on
the big segment ω′ µω which is the complement of the small segment obtained
by reflecting the original one on ω′ ω. If we take z on the semicricle ω′ vω on
ω′ ω as diameter, then z and z∗ would lie on the same segment. So, in this case,
under the same argument, ǫ > 1.
In either case, ǫ > 1 and consequently ǫ ′ < 1 so that ǫ − ǫ ′ > 0. But
ǫ − ǫ ′ = γ(ω − ω′ ) implies that γ > 0.
Now, if z lies on ω′ vω, u is purely imaginary with arg u = π/2. On setting
u = u′ i, we obtain the transformation u′ → u′ ∗ = ǫ 2 u′ ; u′ is real and positive.
Hereafter we need not distinguish between the two cases.
If we define v such that u = ǫ 2v (writing u instead of u′ ) then u → u∗
P
becomes v → v + 1. The function f (z, s) = y s ′ m,n |m + nz|−2s , is invariant
under the modular substitution z → (αz + β)(γz + δ)−1 and hence as a function
of v, it has period 1. It has a valid Fourier expansion in v of the form f (z, s) =
∞
P
ak e2πikv . Now
k=−∞
ak =
Z
1
f (z, s)e−2πikv dv
0
1
=
2 log ǫ
Z
1
ǫ2
f (z, s)u−πik/ log ǫ
du
.
u
On changing the variable from u to z, the series f (z, s) being uniformly
convergent on the corresponding segment of ω′ vω, we may interchange integration and summation and obtain
Z 2
ys
1 X ǫ
du
ak =
u−πik/ log ǫ .
2s
2 log ǫ m,n 1 |m + nz|
u
86
Applications to Algebraic Number Theory
From the relation z =
y=
79
ωui + ω′
it follows that
ui + 1
(m + nω)ui + (m + nω′ )
u(ω − ω′ )
and
m
+
nz
=
ui + 1
u2 + 1
Setting β = m √
+ nω then β ∈ b since m and n are both rational integers and
ω − ω′ = N(b) d. Then our integral becomes
!s
X Z ǫ2
u
1
du
s s/2
(N(b)) d
ak =
u−πik/ log ǫ .
2
2
2
′
2 log ǫ
u
β u +β
b|β,0 1
To bring the right hand-side to proper shape again, we use an idea of Hecke.
We shall now get rid of β in the integrand by introducing U = u|β/β′ |. The
integral then reduces to
Z |(βǫ/β′ ǫ ′ )| s−(πik/ log ǫ)
πik/ log ǫ
u
du
β −s
|N(β)|
,
β′ 2
s
(u + 1) u
|β/β′ |
on writing u instead of U. For two elements β, γ, , 0, (β) = (γ) implies√that
γ = ±βǫ n , n = 0, ±1, ±2, . . . with ǫ > 1, being the fundamental unit in Q( d).
Then
n γ
βǫ
|γ| = |βǫ n |, |γ′ | = |β′ ǫ ′ n | and ′ = ′ ′ n .
γ
βǫ
n
We have therefore, on taking into account γ = βǫ of γ = −βǫ n ,
X Z |(βǫ/β′ ǫ ′ )| u s−(πik/ log ǫ) du
′
(u2 + 1) s u
b|β,0 |β/β |
∞ Z |(βǫ n+1 /β′ ǫ ′ n+1 )| s−(πik/ log ǫ)
X X
u
du
.
=2
2
s
n ′ ′n
(u + 1) u
b|(β),0 n=−∞ |(βǫ /β ǫ )|
Now since ǫ > 1, the intervals (|β/β′ ) · ǫ 2n , |β/β′ | · ǫ 2(n+1) ) fill out the halfline (0, ∞) exactly once, without gaps and overlaps, as n tends to −∞ on one
side and +∞ on the other side. Hence we have
Z ∞ s−(πik/ log ǫ)
∞ Z |(βǫ n+1 /β′ ǫ ′ n+1 )| s−(πik/ log ǫ)
X
u
u
du
du
2
=2
·
· ,
2 + 1) s
2 + 1) s
n
n
′
′
u
u
(u
(u
0
n=−∞ |(βǫ /β ǫ )|
and
2
Z
∞
0
!
!
s
s
πik
πik
Γ
Γ
−
+
2 2 log ǫ
2 2 log ǫ
u s−(πik/ log ǫ) du
=
Γ(s)
(u2 + 1) s u
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Applications to Algebraic Number Theory
80
Therefore, we have
Z 1
ak =
f (z, s)e−2πikv dv
0
!
!
s
πik
πik
s
Γ
−
+
Γ
2 2 log ǫ
2 2 log ǫ
1
(N(b)) s d s/2
×
=
2 log ǫ
Γ(s)
X β πik/ log ǫ
×
|N(β)|−s ,
β′ b|(β),0
since the expression under the summation Σ does not change for associated
elements β and γ.
Upto a product of Γ-factors, the Fourier coefficients ak are Dirichlet series.
For k = 0,
2 s
Z 1
Γ
X
1
2 d s/2
(N(a))−s
a0 =
f (z, s)dv =
2 log ǫ Γ(s)
0
a∈A
2 s
1 Γ 2 s/2
=
d ζ(s, A),
2 log ǫ Γ(s)
where A denotes the ideal class of b−1 in the wide sense. If k , 0, we define
for σ > 1,
X
b
χ(a)(N(a))−s
ζ(s, b
χ, A) =
a∈A
= (b
χ(b))−1
X
a∈A
ab=(β)
b
χ((β))(N(a))−s ;
The function ζ(s, b
χ, A) is called the zeta-function of the class A and associated
with the Grössencharacter b
χ, which is defined as follows:
The character b
χ is defined on principal ideals (β) of K0 as
πik/ log ǫ
β
b
χ((β)) = ′ β
(b
χ((β)) is independent of the generator β by definition). Then b
χ is extended to
all ideals i as follows: If ik = (v), define b
χ(i) as a hth root of b
χ((v)) so that
πik/ log ǫ
v
b
χ(ih ) = b
χ((v)) = ′ v
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Applications to Algebraic Number Theory
It is clear that b
χ is multiplicative.
We have now
b
χ(b)ζ(s, b
χ, A) =
81
89
X
a∈A
ab=(β),(0)
= (N(b)) s
b
χ((β))(N(a))−s
X
b|(β),(0)
b
χ((β))|N(β)|−s ,
so that we may rewrite the expression for ak as follows:
!
!
s
s
πik
πik
Γ
Γ
−
+
2 2 log ǫ
2 log ǫ s/2 2 2 log ǫ
b
ak =
χ(b)ζ(s, b
χ, A).
d
Γ(s)
One can make some applications from the nature of the Fourier coefficients
ak .
We know that the function f (z, s) satisfies the following functional equation, viz.
π−s Γ(s) f (z, s) = π−(1−s) Γ(1 − s) f (z, 1 − s).
One can then show that from the analytic continuation of f (z, s) it follows
that the Fourier coefficients ak as functions of s have also analytic continuations into the whole s-plane and satisfy a functional equation similar to the
above. In the particular case, when k = 0, it follows that
!
s
1−s
π−s d s/2 Γ2
ζ(1 − s, A).
ζ(s, A) = π−(1−s) d(1−s)/2 Γ2
2
2
P
Hence, for the zeta-function ζk0 (s) = ζ(s, A), we have
A
−s s/2 2
π d
Γ
s
2
−(1−s) (1−s)/2 2
ζK0 (s) = π
d
Γ
!
1−s
ζK0 (1 − s).
2
This was discovered first, by Hecke, who also introduced the zeta-functions
with Grössencharacters and derived a functional equation for the same.
We shall now use Kronecker’s first limit formula to study the behaviour of
ζ(s, A) at s = 1.
From Kronecker’s first limit formula, we have
f (z, s) =
π
√
+ 2π(C − log 2 − log y|η(z)|2 + · · · ).
s−1
90
Applications to Algebraic Number Theory
82
It can be shown that the series on the right can be integrated term by term with
respect to v (after transforming z into v) in the interval (0, 1), i.e.
2 s
Z 1
Γ
1
2 d s/2 ζ(s, A)
f (z, s)dv =
2
log
ǫ
Γ(s)
0
Z 1
π
√
+ 2π(C − log 2) − 2π
log( y|η(z)|2 )dv + · · · .
=
s−1
0
It follows therefore that the function on the left has a pole at s = 1 with
residue π and the constant term in the expansion is provided by the integral
which cannot in general be computed. It is to be noted here that in the case of
the imaginary quadratic field, we had only the integrand on the right side and
the argument z was an element of the field, but here z is a (complex) variable.
One cannot get rid of the integral even if one uses the series expansion for the
integrand.
The above formula was found by Hecke and is the Kronecker limit formula for a real quadratic field. One can also consider ak for k , 0 and obtain
a similar formula.
Changing this integral to a contour integral, we shall later obtain an analogue of Kronecker’s solution of Pell’s equation in terms of elliptic functions.
We have now, for real v,
Z 1
Z v+1
a0 =
f (z, s)dv =
f (z, s)dv
0
v
2 s
Γ
1
2 d s/2 ζ(s, A).
=
2 log ǫ Γ(s)
Using the Legendre formula
s s + 1! √
Γ
= π21−s Γ(s),
Γ
2
2
we obtain
2
Γ
Γ(s)
Γ(s)
s =
s
2
Γ2
Γ2
2
2 Γ
But on the other hand,
!
!
s+1
2 s+1
Γ
2
2
! = 2(1−s)
.
π2
Γ(s)
s+1
2
!
s+1
Γ
2
= 1 + terms in (s − 1)2 ,
Γ(s)
2
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Applications to Algebraic Number Theory
83
so that we have
ζ(s, A) = 2 log ǫd−s/2
Γ(s)
s
Γ2
2
Z
v+1
f (z, s)dv
v
1
2 log ǫ
= √ (22 d− 2 )(s−1) (1 + terms in (s − 1)2 )
π d
Z
v+1
f (z, s)dv.
v
√
4
1
By setting (22 d− 2 )(s−1) = e(s−1)(2 log 2−2 log d) and expanding it in powers of (s −
1), and applying Kronecker’s first limit formula for f (z, s), we obtain finally,
ζ(s, A) =
√4
2 log ǫ
√ (1 + (s − 1)(2 log 2 − 2 log d) + · · · )×
π d
√4
1
+ 2C − (2 log 2 − 2 log d)−
×
s−1
!
Z v+1
√ √4
−2
log( y d|η(z)|2 )dv + · · · ,
v
i.e.
1
2 log ǫ
+ 2C − 2
ζ(s, A) = √
s
−
1
d
Z
v+1
v
!
√ √4
2
log( y d|η(z)| )dv + · · · .
(79)
2 log ǫ
√ which
d
is independent of the ideal class A. This result was first discovered by Dirichlet.
It would be nice if one could compute the integral and express it in terms
of an analytic function, but it looks impossible. We shall only simplify it to a
certain extent by converting it into a contour integral.
z − ω′
follows that
From the substitution ui =
ω−z
!
z − ω′
y(ω − ω′ )
u = Im
=
ω−z
(z − ω)(z − ω)
From this, we deduce that ζ(s, A) has a pole at s = 1 with residue
and similarly
u−1 =
On multiplying the two, we have
1= (
y(ω − ω′ )
.
(z − ω′ )(z − ω′ )
′
(z − ω)(z − ω )
ω − ω′
y2
)(
(z − ω)(z − ω′ )
ω − ω′
).
(80)
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Applications to Algebraic Number Theory
84
Consider now the expression
F(z) =
(z − ω)(z − ω′ )
= az2 + bz + c.
N(b)
(81)
We then claim that a, b, c are rational with (a, b, c) = 1 and b2 − 4ac = d. From
Kronecker’s generalization of Gauss’s theorem on the content of a product of
polynomials with algebraic numbers as coefficients, we have for the product
(ξ − ωη)(ξ − ω′ η), (1, −ω)(1, −ω′ ) = (1, −(ω + ω′ ), ωω′ ) or in other words,
bb′ = (1, λ, µ) where λ and µ are rational, i.e.
!
λ
µ
1
= 1.
,
,
N(b) N(b) N(b)
But, from our definition of F(z), a =
1
λ
µ
, b =
, c =
so that
N(b)
N(b)
N(b)
(a, b, c) = 1. The discriminant of F(z) is
b2 − 4ac =
(ω + ω′ )2 − 4ωω′ (ω − ω′ )2
=
= d.
N(b)2
N(b)2
1
> 0. We can then show that the class of the quadratic form
N(b)
F(z) is uniquely determined by the ideal class A.
√ √4
From (80) it follows that y2 d = F(z) · F(z) or equivalently, y d =
p4
F(z) · F(z).
Now, for the integral in (79), we have
Further a =
−4 log ǫ
√
d
Z
v+1
√ √4
log( u d|η(z)|2 )dv = 2
v
since, by definition of v,
Z
z∗
z
p4
dz
log(| F(z)η(z)|2 )
,
F(z)
√
dz
− d
dv =
·
2 log ǫ F(z)
and the transformation v → v + 1 corresponds to z → z∗ on the segment of the
orthogonal circle. On the right side, z may be taken arbitrarily on
√ the segment.
Thus, let A be an ideal class, in the wide sense, of K0 = Q( d), d > 0 and
′
b αanβ ideal in A with an integral basis [1, ω], ω > ω . Corresponding to b, let
∗
−1
γ δ be defined as on p. 85 and let for z ∈ h, z = (αz + β)(γz + δ) . Let,
further, ǫ > 1 be the fundamental unit in K0 . Then we have
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Applications to Algebraic Number Theory
85
Theorem 8 (Hecke). For the zeta-function ζ(s, A) associated with the class A,
we have the limit formula
!
Z z∗
p4
dz
2 log ǫ 1
4C log ǫ
lim ζ(s, A) − √
+2
log(| F(z)η(z)|2 )
,
= √
s→1
s
−
1
F(z)
z
d
d
where the integration is over a segment zz∗ of the semicircle on ω′ ω as diameter
and F(z) is defined by (81).
Now, the question arises whether one could transform the integral in such a
way that it is independent of the choice of z and also the path of integration. It
is possible to do that, by the method of Herglotz who reduced this to an integral
involving simpler functins than η(z).
If N(ǫ) = 1, then the transformation v → v + 1 is equivalent to z → z∗ =
αz + β
with αγ βδ , a modular matrix.
γz + δ
If N(ǫ) = −1, on replacing ǫ by ǫ 2 , N(ǫ 2 ) = 1 and the transformation
′ ′
α′ z + β′
v → v + 1 goes over to the transformation z → ′
with αγ′ βδ′ a modular
′
γ z+δ
matrix. Further, because of the periodicity of f (z, s),
Z
v
v+2
f (z, s)dv = 2
Z
v+1
f (z, s)dv
v
so that we may assume without
√ loss of generality that N(ǫ) = 1.
Now, the behaviour of 4 F(z)√· η(z) under a modular substitution can be
studied. We know that η(z∗ ) = ρ γz + δη(z) where ρ is a 24th root of unity.
Further,
(z − ω)(z − ω′ )
(z∗ − ω)(z∗ − ω′ ) =
(γz + δ)2
if one uses the fact that ω = ω∗ and γω + δ = ǫ.
From these two, it follows that
√4
p4
F(z)
∗
F(z ) = κ √
γz + δ
with κ, a 4th root of unity. Therefore
p4
p4
F(z∗ )η(z∗ ) = ρκ F(z)η(z).
(82)
(We emphasize here that (82) holds only for a hyperbolic substitution and a
power of the same, since we have made essential use of the fact that ω and ω′
94
Applications to Algebraic Number Theory
86
are fixed points of the substitution in obtaining the transformation formula for
F(z).)
√
Consider the function 4 F(z)η(z). It is an analytic function having no zeros in
√ the upper half plane so that on choosing a single valued branch of
log( 4 F(z)η(z)), we have
√ a regular function in the upper z-half plane.
Let us denote log 4 F(z)η(z) by g(v). Then (82) implies that g(v+1)−g(v) =
2πiλ (say) with λ rational. One can then express λ by means of the so-called
Dedekind sums. Setting g(v) − 2πiλv = h(v), we have h(v + 1) = h(v) possesses
∞
P
cn e2πinv , where
a Fourier development in e2πiv , or in other words, h(v) =
n=−∞
R v+1
cn are constants, since h(v) is an analytic function of v and c0 = v h(v)dv.
Here v lies in a certain strip enclosing the real axis.
R v+1
Now, consider the complex integral c0 = v h(v)dv. Here v is, in general,
complex and the path of integration may be any curve between v and v+1 lying
in the strip in which the Fourier expansion is vaid. We have
√ Z z∗
Z v+1
p4
dz
d
−
log(| F(z)η(z)|2 )
=2
Re (g(v))dv (v real)
2 log ǫ z
F(z)
v
!
Z v+1
= 2 Re
h(v)dv
95
v
= 2 Re (c0 ).
The computation of the integral on the left side therefore reduces to the determination of c0 , which in turn is independent of v and the path of integration.
The trick for computing the integral is to convert it into an infinite integral by
letting z → ∞ and z∗ → α/γ. Expanding g(v) in an infinite series, one can
express this infinite integral as a definite integral of an elementary function.
(See G. Herglotz.)
We shall now obtain an analogue of Kronecker’s solution of Pell’s equation
in this case. Let us define
Z z∗
p4
dz
.
(83)
g(A) = 2
log(| F(z)η(z)|2 )
F(z)
z
Then, associated with a character χ of the ideal class group (in the wide sense),
we have
X
L(s, χ) =
χ(A)ζ(s, A),
A
the summation running over all wide classes A.
96
Applications to Algebraic Number Theory
From (79), we obtain, for χ , 1, the expansion,
X
L(s, χ) =
χ(A)g(A) + terms involving (s − 1).
87
(84)
A
Before proceeding to derive the analogue of Kronecker’s solution of Pell’s
equation from the above, we shall examine under what conditions,
√ a genus
character, which is a character of the narrow class group of Q( d), is also
a character of the wide class group. Let χ be a genus character, defined as
follows: If d =
! d1 d2 , then for all ideals a with (a, d1 ) = (1), χ is defined by
d1
. (We may suppose without loss of generality that d1 is odd).
χ(a) =
N(a)
Now, in general, χ is not a character of the wide class group. It will be so, if
χ((α)) = 1 for all integers α with (α, d1 ) = (1). For elements α with N(α) > 0,
this is true by the definition of χ.
If N(α) < 0, then
!
!
d1
d1
=
=1
χ((α)) =
N((α))
−N(α)
for all α, if it is true for one such α.
√
Take α = 1 + d so that N(α) = 1 − d < 0. We need examine only
d1 being odd. Then
!
d1
,
d−1
!
!
! 


d1
d−1
−1
+1 if d1 > 0
=
=
=
−1 if d < 0.

d−1
d1
d1
1
Since d is positive, either both d1 , d2 are positive or both are negative. In
the former case, χ continues to be a character of the wide class group and in
the latter case, it is no longer a character of the wide class group.
Case (i). Let us assume, that both d1 and d2 are positive. Then the genus
character χ as defined above, is also a wide class character.
We have a decomposition of L(s, χ) due to Kronecker, from (66) as follows:
L(s, χ) = Ld1 (s)Ld2 (s) and on taking the values on both sides at s = 1, we obtain
L(1, χ) = Ld1 (1)Ld2 (1).
P
From (84), we have now L(1, χ) = χ(A)g(A), summation running over
A
all wide classes A.
Further
Ld1 (1) =
2 log ǫ1 h1
√
d1
97
Applications to Algebraic Number Theory
88
√
if ǫ1√denotes the fundamental unit of Q( d1 ) and h1 , the wide class number of
Q( d1 ) and similarly for Ld2 (1). Thus as an analogue of Kronecker’s solution
of Pell’s equation as derived in (68), we have the following
√
Proposition 13. Let χ be a genus character of Q( d), d > 0, corresponding
to the decomposition d = d1 d2 (d1 > 0, d2 > 0) of√d. Let h√1 , h2 be the class
numbers and ǫ1 , ǫ2 the fundamental units of Q( d1 ), Q( d2 ) respectively.
Then
√ X
χ(A)g(A),
(85)
4h1 h2 log ǫ1 log ǫ2 = d
A
√
where A runs over all the ideal classes of Q( d) in the wide sense and g(A) is
defined by (83).
The expression on the right side of (85) is not very simple. It is not known
whether the number on the left side is rational or irrational. It is probable that
the number on the left side is a complicated transcendental number, so that one
cannot expect a simple value on the right side.
Case (ii). Suppose d1 and d2 are both negative. Then again from the decomposition formula (66), we have
4π2 h1 h2
2πh2
2πh1
=
√
√
√
w1 |d1 | w2 |d2 | w1 w2 d
√
√
where h1 , h2 denote the wide class numbers
of Q(√ d1 ) and Q( d2 ), and w1 ,
√
w2 the number of roots of unity in Q( d1 ) and Q( d2 ) respectively.
We shall see in § 5 that
L(1, χ) = Ld1 (1)Ld2 (1) =
π2 X
χ(B)G(B),
L(1, χ) = √
d B
√
B running over all narrow classes of Q( d) and G(B) being numbers depending only on B.
We shall further prove that G(B) are rational numbers which can be realized
in terms of periods of certain abelian integrals of the third kind.
We then have, as an analogue of Kronecker’s solution of Pell’s equation,
the following:
4h1 h2 X
=
χ(B)G(B).
w1 w2
B
√
Example. Consider the field Q( 10) with discriminant d = 40 = 5.8; d1 = 5
and d2 = 8.
98
Applications to Algebraic Number Theory
89
√
√
The class number of Q( 10) is 2. Since the fundamental unit ǫ = 3 + 10
has norm −1, the narrow classes are the same as wide classes. We can now
choose the two ideal class representatives as follows
"
#
√
1√
1
10 with N(b2 ) = .
(1) = b1 = [1, 10] and b2 = 1,
2
2
√
√
The class numbers h1 and h2 of Q(√ 5) and Q( 8) are both 1 and the funda√
1+ 5
and 1 + 2. The only character χ , 1
mental units are respectively
2
has the property that χ(b1 ) = 1 and χ(b2 ) = −1, so that we have from (85), the
following:
√ 

√
√
 1 + 5 
 · log(1 + 2) = 10(g(E) − g(A)).
2 log 
2
We shall make a remark
for more general applications. Consider the abso√
lute class field of Q( d) with d > 0. For computing the class number of this
field, one can proceed in the same way, as in the case of an imaginary quadratic
field. For the same, one requires the computation of L(1, χ) for arbitrary narrow
class characters. This will be done in § 5, for a special type of characters.
√
4 Ray class fields over Q( d), d < 0
99
Let K be an algebraic number field of degree n over Q, the field of rational
numbers. Let r1 and 2r2 be the number of real and complex conjugates of K
respectively, so that r1 + 2r2 = n. Let K (1) , . . . , K (r1 ) be the real conjugates
of K and K (r1 +1) , . . . , K (n) be the complex conjugates of K. Let, for α ∈ K,
α(i) ∈ K (i) , i = 1, 2, . . . , n, denote the conjugates of α. Further let f be a given
α2
α1
, γ2 =
with α1 , β1 , α2 , β2 integral
integral ideal in K. Two numbers γ1 =
β1
β2
in K and with β1 β2 coprime to f are (multiplicatively) congruent modulo f (in
symbols, γ1 ≡ γ2 ( mod ∗ f) if α1 β2 ≡ α2 β1 ( mod f)). If γ1 , γ2 are integers in
K, this is the usual congruence modulo f.
b
Let us consider non-zero fractional ideals a of the form a = where b and
c
c are integral ideals in K coprime to f. These fractional ideals a form, under the
usual multiplication of ideals, an abelian group which we shall denote by Gf .
Let Gf be the subgroup of Gf consisting of all principal ideals (α) for which
α > 0 (i.e. α(i) > 0, for i = 1, 2, . . . , r1 ) and α ≡ 1( mod ∗ f). It is clear that if
(γ) ∈ Gf and γ ≡ 1( mod ∗ f), then γ might be written as α/β where α and β
Applications to Algebraic Number Theory
90
are integers in K satisfying the conditions, α > 0, β > 0, α ≡ 1( mod f) and
β ≡ 1( mod f).
The quotient group Gf /Gf is a finite abelian group called the ray class group
modulo f; its elements are called ray classes modulo f. Two ideals a and b in Gf
are equivalent modulo f(a ∼ b mod f) if they lie in the same ray class modulo
f, i.e. a = (γ)b with (γ) ∈ Gf .
If r1 = 0 and f = (1), equivalence modulo f is the usual equivalence in the
wide sense and the ray class group modulo f is the class group of K in the wide
sense. If r1 = 2, r2 = 0 (i.e. K is a real quadratic field) and f = (1), then the
equivalence modulo f is the equivalence in the narrow sense and the ray class
group is the class group of K in the narrow sense. Let χ be a character of the
group of ray classes modulo f. Then associated with χ, we define for σ > 1,
the L-series
X
L(s, χ) =
χ(a)(N(a))−s
a,0
where N(a) is the norm of a in K and the summation is extended over all integral ideals a coprime to f. It is clear that L(s, χ) is a regular function of s for
σ > 1. Moreover, due to the multiplicative character of χ, we have for σ > 1,
an Euler-product decomposition for L(s, χ), namely
Y
L(s, χ) =
(1 − χ(p)(N(p))−s )−1
p|f
where the infinite product is extended over all prime ideals p coprime to f.
Hecke has shown that L(s, χ) can be continued analytically as a meromorphic function of s in the whole s-plane and that when χ is a “proper” character,
there is a functional equation relating L(s, χ) with L(1 − s, χ), where χ is the
conjugate character. If χ , 1 (the principal character), then L(s, χ) is an entire
function of s. If χ = 1 and f = (1), then L(s, χ) is the Dedekind zeta function
of K.
We are interested in determining the value at s = 1 of L(s, χ), in the case
when K is a real or imaginary quadratic number field and χ is not the principal
character. For this purpose, we need to apply Kronecker’s second limit formula. Later, we shall use this for the determination of the class number of the
“ray class field” of K, corresponding to the ideal f.
First we shall investigate the structure of a ray class character χ. Let α and
α2
β(, 0) be integers coprime to f such that α = β( mod f). Then γ = 2 satisfies
β
γ = 1( mod ∗ f) and γ > 0 so that χ((γ)) = 1 i.e. χ((α2 )) = χ((β2 )). In other
words,
χ((α)) = ±χ((β)) or χ((α/β)) = ±1.
100
Applications to Algebraic Number Theory
91
Let L be the multiplicative group of λ , 0 in K. Corresponding to a ray
class character χ modulo f, we define a character v of L as follows. For λ ∈ L,
we find an integer α ∈ K such that α ≡ 1( mod f) and αλ > 0 and define
v(λ) = χ((α)). It is first clear that v(λ) is well-defined; for, if v(λ) = χ((β))
for another integer β ∈ K satisfying the same conditions, then we see at once
α
α
≡ 1( mod ∗ f) and
> 0 and so χ((α)) = χ((β)). It is easily verified
that
β
β
that v(λµ) = v(λ)v(µ). Moreover, v(λ) = ±1; for, λ2 > 0 and by definition,
v(λ2 ) = χ((α)) for α satisfying α ≡ 1( mod f) and α > 0 so that (v(λ))2 =
v(λ2 ) = χ((α)) = +1. We shall call v(λ), a character of signature.
Consider the subgroup B of L consisting of λ > 0. Clearly v(λ) = 1 for all
λ ∈ B. Thus v(λ) may be regarded as a character of the quotient group L/B.
Now L/B is an abelian group of order 2r1 exactly, since one can find λ ∈ L for
which the real conjugates λ(i) , i = 1, 2, . . . , r1 , have arbitrarily prescribed signs.
Also, there are 2r1 distinct characters u of L/B defined by
!g
r1
Y
λ(i) i
u(λ) =
,
|λ(i) |
i=1
gi = 0 or 1, λ ∈ L.
There can indeed be no more that 2r1 characters of L/B and hence v(λ) coincides with one of these characters u, of L/B. We do not however assert that
every character u of L/B is realizable from a ray class character modulo f, in
the manner described above.
Let now γ ∈ K such that γ ≡ 1( mod ∗ f) and (γ) ∈ Gf . By definition,
v(γ) = χ((δ)) for an integer δ such that δ ≡ 1( mod f) and δγ > 0. But
χ((δ)) = χ((γ)) and hence v(γ) = χ((γ)). Let α and β(, 0) be two integers
coprime to f such that α = β( mod f). Then γ = α/β ≡ 1( mod ∗ f) and
v(α)
χ((α))
= χ((γ)) = v(γ) =
.
χ((β))
v(β)
Thus for integral α(, 0) coprime to f, the ratio χ((α))/v(α) depends only on
the residue class of α modulo f and is in fact, a character of the group G(f) of
prime residue classes modulo f. We may denote it by χ(α). Thus
Proposition 14. Any character χ((α)) of the ray class group modulo f may be
written in the form
χ((α)) = v(α)χ(α)
where v(α) is a character of signature and χ(α) is a character of the group
G(f).
101
Applications to Algebraic Number Theory
92
Before proceeding further, we give a simple illustration of the above. Let
us take K to be Q, the field of rational numbers and f to be the principal ideal
(|d|) in the ring of rational integers, d being the discriminant of a quadratic field
over Q. For an integer m coprime to d, we define
!


d m



for d > 0


 |m| ! |m|
χ((m)) = 

d



for d < 0

 |m|
!
d
is the Legendre-Jacobi-Kronecker symbol. The group Gf now
where
|m|
consists of fractional ideals (m/n) where m and n are rational integers coprime
to d. We extend χ to the ideals (m/n) in Gf by setting χ((m/n)) = χ((m))/χ((n)).
Now it is known that if m ≡ n( mod d) and mn is coprime to d, then
!
!
d
d
=
if d > 0
|m|
|n|
and
102
!
!
d n
d m
=
if d < 0
|m| |m|
|n| |n|
Thus
d
χ(m) =
|m|
and
χ(m) =
!
for
!
d m
|m| |m|
d>0
for
d<0
m
is clearly a character of signa|m|
ture. It is easily verified that χ((m/n)) is a ray class character modulo (|d|) and
that χ((m)) = v(m)χ(m).
PP
−s
Now L(s, χ) =
a (a)(N(a)) , where A runs over all the ideal classes
are characters of G(|d|). Moreover v(m) =
A
in the wide sense and a over all the non-zero integral ideals in A, which are
coprime to f. In the class A−1 , we can choose an integral ideal bA coprime
to f and abA = (β) where β is an integer divisible by bA and coprime to f.
Conversely, and principal ideal (β) divisible by bA and coprime to f is of the
form abA , where a is an integral ideal in A coprime to f. Moreover, χ(bA )χ(a) =
χ((β)) and N(bA ) · N(a) = |N(β)|, where N(β) is the norm of β. Thus
X
X
L(s, χ) =
χ(bA )(N(bA )) s
χ((β))|N(β)|−s
A
bA |(β)
103
Applications to Algebraic Number Theory
93
where the inner summation is over all principal ideals (β) divisible by bA and
coprime to f. Since A−1 runs over all classes in the wide sense when A does
so, we may assume that bA is an integral ideal in A coprime to f. Moreover, in
view of Proposition 14, we have for σ > 1,
X
X
L(s, χ) =
χ(bA )(N(bA )) s
v(β)χ(β)|N(β)|−s
(86)
A
bA |(β)
where now A runs over all the ideal classes of K in the wide sense, bA is a fixed
integral ideal in A coprime to f and the inner sum is over all principal ideals
(β) divisible by bA and coprime to f. We may extend χ(β) to all residue classes
modulo f by setting χ(α) = 0 for α not coprime to f. Thus we may regard the
inner sum in (86) as extended over all principal ideals (β) divisible by bA . In
order to render the series in (86) suitable for the application of Kronecker’s
limit formula, we have to replace χ(β) by an exponential of the form e2πi(mu+nv)
occurring in the limit formula. We shall, in the sequel, express χ(β) as an
exponential sum by using an idea due to Lagrange, which is as follows.
Let x1 , . . . , xn be n distinct roots of a polynomial f (x) of degree n, with
coefficients in a field K0 containing all the nth roots of unity. Let, further, the
field M = K0 (x1 , . . . , xn ) be an abelian extension of K0 , with galois group H.
Let us assume moreover that if σ1 , . . . , σn ∈ H, then xi = x1σi i = 1, 2, . . . , n.
n
P
Now, if χ is a character of H, then let us define yχ = χ(σi )xi . It is clear that
i=1
yχ ∈ M and
σ
yχ j =
n
X
σ σj
χ(σi )x1 i
i=1
= χ(σ j )
n
X
σ σj
χ(σi σ j )x1 i
= χ(σ j )yχ .
i=1
Hence ynχ ∈ K0 . Now, if χ denotes the conjugate character of χ and σ ∈ H, then
yχ =
n
X
χ(σi σ)x1σi σ = χ(σ)
i=1
If yχ , 0, then χ(σ) = y−1
χ
n
P
i=1
n
X
χ(σi )xiσ .
i=1
χ(σi )xiσ . We shall use a similar method to express
χ(β) as an exponential sum whose terms involve β in the exponent.
First we need the following facts concerning the different of an algebraic
number field M of finite degree over Q. Let for α ∈ M, S (α) denote the
trace of α. Let a be an ideal (not necessarily integral) in M and let a∗ be the
“complementary” ideal to a, namely the set of λ ∈ M, for which S (λα) is a
rational integer for all α ∈ a. It is known that aa∗ is independent of a and in fact
104
Applications to Algebraic Number Theory
94
aa∗ = (1)∗ = ϑ−1 , where ϑ is the different of M. Further clearly (a∗ )∗ = a and
N(ϑ) = |d|, where d is the discriminant of M. These facts can be verified in a
simple way if M is a quadratic field over Q, with discriminant d. Let [α1 , α2 ]
be an integral basis of a. The ideal a∗ is the set of λ ∈ M for which S (λα1 )
and S (λα2 ) are both rational integers. If α′1 , α′2"denote the conjugates of α1#
α′2
−α′1
and α2 respectively, then it is easliy verifed that
,
α1 α′2 − α2 α′1 α1 α′2 − α2 α′1
√
′
∗
′
is an integral basis of a . Now α1 α2 − α2 α1 = ±N(a) d and this means that
√
√
a∗ =√(1/N(a) d)a′ . Since (N(a)) = aa′ , we see that a∗ = a−1 (1/ d) i.e. aa∗ =
(1/ d) = ϑ−1 . Moreover N(ϑ) = |d|.
Let us consider now the ideal a = f−1 ϑ−1 in K; clearly a∗ = f. Let us choose
in the class of fϑ, an integral ideal q coprime to f. Then qf−1 ϑ−1 = (γ) for γ ∈ K
i.e. (γ)ϑ = qf−1 has exact denominator f. If K were a quadratic field and f, a
principal ideal, then fϑ is principal and we may take q = (1). We shall consider
γ fixed this way once for all, in the sequel. Let us observe that if λ ∈ f, S (λγ)
is a rational integer.
With a view to express χ(β) as an exponential sum, we now define the sum
X
χ(λ)e2πiS (λγ)
(87)
T=
λ
mod f
where λ runs over a full system of representatives of residue classes modulo
f; for f = (1), clearly T = 1. We see that T is defined independently of the
choice of representatives λ, for, if µ runs over another system of representatives
modulo f, then λ ≡ µ( mod f) in some order and in this case, χ(λ) = χ(µ) and
e2πiS (λγ) = e2πiS (µγ) since S ((λ−µ)γ) is a rational integer. Moreover, in this sum,
λ may be supposed to run only over representatives of elements of G(f), since
χ(α) = 0, for α not coprime to f. The sums of this type were first investigated
by Hecke; similar sums for the case of the rational number field have been
studied by Gauss, Dirichlet and for complex characters χ, by Hasse.
Let α be an integer in K coprime to f. Then αλ runs over a complete set of
prime residue classes modulo f when λ λ does so. Thus
X
χ(αλ)e2πiS (αλγ)
T=
λ mod f
χ(λ)e2πiS (αλγ) ,
λ
X
χ(λ)e2πiS (αλγ) .
λ
X
= χ(α)
i.e.
T χ(α) =
mod f
mod f
(88)
105
Applications to Algebraic Number Theory
95
We shall presently see that (88) is true even for α not coprime to f and that
T , 0, when χ is a so-called proper character of G(f).
Let g be a proper divisor of f and ψ, a character of G(g). We may extend
ψ into a character χ of G(f) by setting χ(λ) = ψ(λ) for λ coprime to f and
χ(λ) = 0, otherwise. We say that a character χ of G(f) is proper, if it is not
derivable in this way from a character of G(g) for a proper divisor g of f.
A ray class character χ modulo f is said to be proper if the associated character χ of G(f) is proper; χ is then said to have f as conductor.
Let χ be a ray class character modulo f and let g be a proper divisor of
f. Moreover let for integers α, β coprime to f such that α ≡ β( mod g) and
αβ > 0, χ(α) = χ(β). We can associate with χ, a ray class character χ0 modulo
g as follows. If p is a prime ideal coprime to f, define χ0 (p) = χ(p). If p is a
prime ideal coprime to g but not to f, we can find a number α such that α ≡ 1(
mod ∗ g), α > 0 and (α)p is coprime to f. We then set χ0 (p) = χ((α)p). We
see that χ0 is well-defined for all prime ideals coprime to g and we extend χ0
multiplicatively to all ideals in the ray classes modulo g. Clearly, χ0 is a ray
class character modulo G and for integral α coprime to g, χ0 ((α)) = χ0 (α)v(α),
where χ0 (α) is the associated character of G(g) and v(α) is the same signature
character as the one associated with χ. We now say the ray class character
χ modulo f having the property described above with respect to g, has g as
conductor, if the associated χ0 has g as conductor. Let from now on, χ be a
proper ray class character modulo f. Since χ(α) = 0 for α not coprime to f, all
we need to prove (88) for proper χ and for α not coprime to f is to show that the
right hand side of (88) is zero. Let b be the greatest common divisor of (α) and
f and let g = fb−1 . Since χ is proper, it is not derivable from any character of
G(g). In other words, there exist integers λ and µ coprime to f such that λ ≡ µ(
mod g) and χ(λ) , χ(µ). Otherwise, if for all integers λ and µ coprime to f
and for which λ ≡ µ( mod g) it is true that χ(λ) = χ(µ), then we can define
a character χ0 of G(g) such that for λ coprime to f, χ(λ) = χ0 (λ) which is a
contradiction. Thus we can find integers µ and ν coprime to f such that µ ≡ v(
mod g) and χ(µ) , χ(v). Now clearly

P

χ(µ)
χ(λ)e2πiS (αλγµ)


X


λ mod f
2πiS (αλγ)
χ(λ)e
=
(89)
P


χ(λ)e2πiS (αλγν) .

χ(v)
λ
mod f
λ
mod f
Further since αµ ≡ αν( mod f), S (αλγ(µ − ν)) is a rational integer and hence
X
X
χ(λ)e2πiS (αλγµ) =
χ(λ)e2πiS (αλγν) .
λ
mod f
λ
mod f
106
Applications to Algebraic Number Theory
96
But since χ(µ) , χ(ν), we see from (89), that
X
χ(λ)e2πiS (αλγ) = 0.
λ
mod f
Hence (88) is true for all integral α.
We now proceed to prove the T , 0. In fact, using (89), we have
X
TT = T
χ(α)e−2πiS (αγ)
α
=
mod f
X
χ(λ)
X
e2πiS (α(λ−1)γ)
(90)
α mod f
λ mod f
Now if α runs over a complete system of representatives of residue classes
modulo f, so does α + ν for every integer ν and hence
X
X
e2πiS (µαγ)
e2πiS (µαγ) = e2πiS (µγν)
α
α
mod f
mod f
Thus, if there exists at least one integer ν such that S (µγν) is not a rational
P 2πiS (µαγ)
e
= 0. Now S (µγν) is a rational integer for all integers ν
integer,
α mod f
if and only if µγ ∈ ϑ−1 i.e. if and only if (µ)qf−1 is integral i.e. µ ∈ f. Thus


X

if µ < f,
0,
2πiS (µαγ)
e
=

N(f), if µ ∈ f.
α mod f
√
From this and from (90), we have then |T |2 = N(f), i.e. |T | = N(f). The
determination of the exact value of T/|T | is of the same order of difficulty as
the corresponding problem for “generalized Gauss sums” considered by Hasse.
We have, finally, for proper χ modulo f, as a consequence of (88),
X
χ(λ)e2πiS (λβγ)
(91)
χ(β) = T −1
λ
mod f
Let us notice that β appears in the exponent in the sum on the right-hand side
of (??).
We now insert the value of χ(β) as given by (??) in the series on the right
hand side of (86). In view of the absolute convergence of the series for σ > 1
and in view of the fact that the sum in (93) is a finite sum, we are allowed to
rearrange the terms as we like. We then obtain for σ > 1 and for a ray class
character χ modulo f with f as conductor,
X
1 X
χ(λ)
L(s, χ) =
χ(bA )(N(bA )) s ×
T λ mod f
A
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Applications to Algebraic Number Theory
×
X
bA |(β)
97
v(β)e2πiS (λβγ) |N(β)|−s .
(92)
In (92), λ runs over a full system of representatives of prime residue classes
modulo f, A over representatives of the ideal classes of K in the wide sense and
the inner sum is over all principal non-zero ideals (β) divisible by bA .
We shall use (92) to determine the value of L(s, χ) at s = 1 when K is a
real or imaginary quadratic field over Q. In this section, we shall consider only
the case when K is an imaginary quadratic field over Q, with discriminant
D < 0. Here v(β) = 1 identically, since there is no nontrivial character of
signature. Moreover, we could assume f , (1), for otherwise, L(s, χ) precisely
the Dedekind zeta function of K. Let, therefore, f , (1) and let w and wf denote
respectively the number of all roots of unity and of roots of unity ǫ satisfying
ǫ ≡ 1( mod f). From (92), we have for σ > 1,
L(s, χ) =
X
X
1
χ(λ)
χ(bA )(N(bA )) s ×
w · T λ mod f
A
X
×
e2πiS (λβγ) (N(β))−s
108
(93)
bA |β,0
where the inner summation is over all β , 0 in bA .
We now contend that as λ runs over a full system of representatives of the
prime residue classes modulo f and bA over a complete set of representatives
(integral and coprime to f) of the classes in the wide sense, then (λ)bA covers
exactly w/wf times, a complete system of representatives of the ray classes
modulo f. In fact, let ǫ1 , . . . , ǫv (v = w/wf ) be a complete system of roots of unity
in K, incongruent modulo f. The corresponding residue classes modulo f form
a subgroup E(f) of G(f). Let λi , i = 1, . . . , r be a set of integers whose residue
classes modulo f constitute a full system of representatives of the cosets of
G(f) modulo E(f). Then it is easily verified that when λ runs over the elements
λ1 , . . . , λr and bA over the representatives of the classes in the wide sense, (λ)bA
covers exactly once a full system of representatives of the ray classes modulo
f. Our assertion above is an immediate consequence. Thus, we have from (93),
for σ > 1,
L(s, χ) =
X
1 X
χ(bB )(N(bB )) s
e2πiS (βγ) (N(β))−s ,
T wf B
b |β,0
(94)
B
where B runs over the ray classes modulo f and bB is a fixed integral ideal in B
coprime to f.
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Applications to Algebraic Number Theory
98
Let [β1 , β2 ] be an integral basis of bB ; we can assume, without loss of genβ2
= zB = xB + iyB with yB > 0. Then if β ∈ bB , β = mβ1 + nβ2 for
erality that
β1
rational integers m, n and N(β) = N(β1 ) × |m + nzB |2 . Moreover,
p
N(bB ) |D| = |β1 β2 − β2 β1 | = 2yB N(β1 ).
Thus
X
(N(bB )) s
e2πiS (βγ) (N(β))−s
bB |β,0
!s X
′ 2πiS ((mβ +nβ )γ)
1
2
2yB
= √
|D|
e
m,n
|m + nzB |−2s ,
where, on the right hand side, the summation is over all ordered pairs of rational
integers (m, n) not equal to (0, 0). Let us set now uB = S (β1 γ) and vB = S (β2 γ)
and let f be the smallest positive rational integer divisible by f. Then in view
of the fact that (γ)ϑ has exact denominator f and bB is coprime to f, it follows
that uB and vB are rational numbers with the reduced common denominator f .
Since f , (1), uB and vB are not simultaneously integral. We then have
X
(N(bB )) s
e2πiS (βγ) (N(β))−s
bB |β,0
!s
yBs
2
= √
|D|
X′
m,n
e2πi(muB +nvB ) |m + nzB |−2s .
(95)
We know that the function of s defined for σ > 1 by the infinite series on the
right hand side of (95) has an analytic continuation which is an entire function
of s. Its value at s = 1 is given precisely by Kronecker’s second limit formula.
Indeed, by (39), we have
yB
′
X
m,n
2πi(muB +nvB )
e
−2
|m + nzB |
ϑ1 (vB − uB ZB , zB ) πiu2B zB 2
.
e
= −π = −π log η(zB )
Inserting the factor e−πiuB vB of absolute value 1 on the right hand side, we have
X′
e2πi(muB +nvB ) |m + nzB |−2
yB
m,n
ϑ1 (vB − uB zB , zB ) πiuB (uB zB −vB ) 2
.
e
= −π log η(zB )
(96)
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Applications to Algebraic Number Theory
99
Now we define for real numbers u, v not simultaneously integral and z ∈ H,
the function
ϑ1 (v − uz, z)
ϕ(v, u, z) = eπiu(uz−v)
.
η(z)
In each ray class B modulo f, we had chosen a fixed integral ideal bB with
β2
integral basis [β1 , β2 ] and defined uB = S (β1 γ), vB = S (β2 γ) and zB =
.
β1
Now, the left hand side of (95) depends only on the ray class B. Hence from
(95) and (96), it is clear that log |ϕ(vB , uB , zB )|2 depends only on B and not on
the special choice of bB or β1 , β2 .
From (94), (95) and (96) we can now deduce
Theorem 9. √If χ is a proper ray class character modulo an integral ideal
f , (1) of Q( D), D < 0, the associated L(s, χ) can be continued analytically
into an entire function of s and its value at s = 1 is given by
X
2π
χ(bB ) log |ϕ(vB , uB , zB )|2 ,
(97)
L(1, χ) = −
√
T wf |D| B
where B runs over all the ray classes modulo f and T is the sum defined by
(87).
We shall need (97)√ later for the determination of the class number of the
‘ray class field’ of Q( D).
By the ray class field (modulo f) of an algebraic number field k, we mean
the relative abelina extension K0 of k, with Galois group isomorphic to the
group of ray classes (modulo f) in k such that the prime divisors of f are the
only prime ideals which are ramified in K0 .
We are now interested first in determining the nature of the numbers
ϕ(vB , uB , zB ). For this purpose, we observe that ϕ(v, u, z) is a regular function of
z in H and we shall study its behaviour when z is subjected to modular transformations and the real variables v and u undergo certain linear transformations.
In fact, using the transformation formula for ϑ1 (w, z) and η(z) proved earlier
and the definition of ϕ(v, u, z), one easily verifies the following formulae, viz.
ϕ(v + i, u, z) = −e−πiu ϕ(v, u, z),
ϕ(v, u + 1, z) = −eπiv ϕ(v, u, z),
ϕ(v + u, u, z + 1) = eπi/6 ϕ(v, u, z),
(98)
ϕ(−u, v, −z−1 ) = e−πi/2 ϕ(v, u, z).
Now, corresponding to a modular transformation z → z∗ = (az + b) · (cz + d)−1
we define v∗ = av + bu and u∗ = cv + du. The last two transformation formulae
111
Applications to Algebraic Number Theory
100
for ϕ(v, u, z) above, merely mean that corresponding to the elementary modular
transformations z → z∗ = z + 1 or z → z∗ = −z−1 , we have
ϕ(v∗ , u∗ , z∗ ) = ρϕ(v, u, z),
where ρ is a 12th root of unity. Since these two transformations generate the
modular group, we see that for any modular transformation z → z∗ ,
ϕ(v∗ , u∗ , z∗ ) = ρϕ(v, u, z)
(98)′
where ρ = ρ(a, b, c, d) is a 12th root of unity which can be determined explicitly.
Let (v, u) be a pair of rational numbers with reduced common denominator
f > 1. Let us define for z ∈ H,
Φ(v, u, z) = ϕ12 f (v, u, z).
Then as a consequence of the above formulae for ϕ(v, u, z), we see that Φ(v +
1, u, z) = Φ(v, u, z), Φ(v, u + 1, z) = Φ(v, u, z) and Φ(v∗ , u∗ , z∗ ) = Φ(v, u, z). If
z → z∗ = (az + b)(cz + d)−1 is a modular transformation of level f , then v∗ − v
and u∗ − u are rational integers and in view of the periodicity of Φ(v, u, z) in v
and u, we see that
Φ(v, u, z) = Φ(v∗ , u∗ , z∗ ) = Φ(v, u, z∗ ).
112
Let (vi , ui )i = 1, 2, . . . , q run over all the paits of rational numbers lying
between 0 and 1 and having reduced common denominator f . Then corresponding to each pair (vi , ui ), we have a function Φi (z) = Φ(vi , ui , z) which, as
seen above, is invariant under modular transformations of level f . Moreover,
if z → z∗ = (az + b)(cz + d)−1 is an arbitrary modular transformation, then for
some i, v∗j ≡ vi ( mod 1) and u∗j ≡ ui ( mod 1) so that in view of the periodicity
of Φ(vi , ui , z) in vi and ui we see that
Φi (z∗ ) = Φ(vi , ui , z∗ ) = Φ(v∗j , u∗j , z∗ ) = Φ(v j , u j , z) = Φ j (z).
In other words, the functions Φi (z) are permuted among themselves by an arbitrary modular transformation.
Now, Φi (z) is regular in H and invariant under modular transformations
of level f . Moreover, Φi (z) has in the local uniformizer e2πiz/ f at infinity, a
power-series expansion with at most a finite number of negative powers. Since
Φi (z∗ ) = Φ j (z) for some j, we see that Φi (z) has at most a pole in the local uniformizers at the ‘parabolic cusps’ of the corresponding fundamental domain.
Applications to Algebraic Number Theory
101
Thus Φi (z) is a modular function of level f . Since a modular transformation
merely permutes the functions Φi (z), we see that any elementary symmetric
function of the functions Φi (z) is a modular function. From the theory of complex multiplications, it can be shown by considering the expansions ‘at infinity’
of the functions Φi (z), that Φi (z) satisfies a polynomial equation whose coefficients are polynomials, with rational integral coefficients, in j(z) (the elliptic
modular invariant). If z lies in an imaginary quadratic field K over Q, it is
known that for z < Q, j(z) is an algebraic integer. Since Φi (z) depends integrally on j(z), it follows that for such z, Φi (z) is an algebraic integer. Actually,
from the theory of complex multiplication, one may show that for z ∈ K(z < Q),
Φi (z) is an algebraic integer in the ray class field modulo f over K. In particular,
the numbers ϕ12 f (vB , uB , zB ) corresponding to the ray classes B modulo f in K,
are algebraic integers in the ray class field modulo f, over K. √
Let now K0 be the ray class field modulo f over K(= Q( D), D < 0). Let
∆, R, W and g be respectively the discriminant of K0 relative to Q, the regulator
of K0 , the number of roots of unity in K0 and the order of the Galois group of
K0 over K. Let H and h denote respectively the class numbers of K0 and K. It is
clear that K0 has no real conjugates over Q and has 2g complex conjugates over
Q. Moreover |∆| = |D|g N(ϑ), where N(ϑ) is the norm in K/Q of the relative
discriminant ϑ of K0 over K.
From class field theory, we know that
Y
ζK0 (s) = ζK (s)
L(s, χ0 )
(∗)
χ0 ,1
where χ runs over all the non-principal ray class characters modulo f and if
fχ is the conductor of χ, then χ0 is the proper ray class character modulo fχ
Q
fχ . Multiplying both sides of (∗)
associated with χ. It is known that ϑ =
χ
above by s − 1 and letting s tend to 1, we have
(2π)g · H · R
2πh Y
= √
L(1, χ0 ).
√
W |∆|
w |D| χ,1
But, from (97), we have
L(1, χ0 ) = −
X
2
2π
χ0 (bB0 ) log ϕ vB0 , uB0 , zB0 √
T 0 wfχ |D| B0
where
T 0 = T 0 (χ0 ) =
X
λ mod fχ
χ0 (λ)e2πiS (λγ0 ) ,
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Applications to Algebraic Number Theory
102
√
γ0 ∈ K is chosen such that (γ0 d) has exact denominator fχ , B0 runs over
all the ray classes modulo fχ , bB0 is a fixed integral ideal in B0 coprime to fχ
and wfχ is the number of roots of unity congruent to 1 modulo fχ . Witn N(fχ )
Q
denoting the norm in K over Q, of the ideal fχ , we have ( N(fχ ))|D|g = |∆|. It
χ
is now easy to deduce
Theorem
√ 10. For the class number H of the ray class field modulo f over
K = Q( D), D < 0, we have the formula

 p
N(Fχ ) X
2 
H
W Y 
χ (bB ) log ϕ vB0 , uB0 , zB0 
=
−
h
w · R χ,1  T 0 wfχ B 0 0
0
When f is a rational integral ideal in K, Fueter has derived a ‘similar’ formula for the class number of the ray class field modulo f or the ‘ring class field’
modulo f over K, by using a “generalization” of Kronecker’s first limit formula.
Unlike in the case of the absolute class field over K, it is not possible, in
Q
occurring in the formula for H/h
general, (for f , (1)) to write the product
χ,1
above, as a (g − 1)-rowed determinant whose elements are of the form log |ηi(k) |
where ηi(k) , i = 1, 2, . . . , g − 1 are conjugates of independent units ηi lying in
K0 .
√
5 Ray class fields over Q( D), D > 0.
Let K be a real quadratic field over Q, with discriminant D > 0. Let f be a
given integral ideal in K and χ(, 1), a proper character of the group of ray
classes modulo f in K. As in the case of the imaginary quadratic field, we shall
first determine the value at s = 1 of the L-series L(s, χ) associated with χ and
use this later to determine the class number of special abelian extensions of K.
We start from formula (92) proved earlier, namely, for σ > 1,
X
X
L(s, χ) = T −1
χ(λ)
χ(bA )×
λ mod f
× (N(bA )) s
X
A
bA |(β),(0)
v(β)e2πiS (λβγ) |N(β)|−s ,
where A runs over all the ideal-classes of K in the wide sense, bA is a fixed
integral ideal in A, chosen once for all and coprime to f and the inner sum is
extended over all non-zero principal ideals (β) divisible by bA . Further, χ(λ)
and v(λ) are respectively the character of G(f) and the character of signature
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Applications to Algebraic Number Theory
103
associated with the ray class character χ as in Proposition 14. Besides, γ is a
√
q
number in K such that (γ C) has exact denominator f i.e. (γ) = √ where
f( D)
(q, f) = (1); the number γ will be fixed throughout this section.
Let, for α ∈ K, α′ denote its conjugate. We may, in the above, suppose
β2
, then, without
that [β1 , β2 ] is a fixed integral basis of bA such that if ω =
β1
√
loss of generality, ω > ω′ . We have, then, (ω − ω)|N(β1 )| = N(bA ) · D.
Further, let uA = S (λβ1 γ) and vA = S (λβ2 γ); then, if β = mβ1 + nβ2 , we have
e2πiS (λβγ) = e2πi(muA +nvA ) .
The character of signature v(λ) associated with the ray class character χ
may, in general, be one of the following, namely, for λ , 0 in K,
(i) v(λ) = 1
(ii) v(λ) =
N(λ)
|N(λ)|
(iii) v(λ) =
λ
|λ|
(iv) v(λ) =
λ′
.
|λ′ |
In what follows, we shall be concerned only with such ray class characters χ,
for which the corresponding v(λ) is defined by (i) or (ii). We shall not deal
with the characters χ, for which either (iii) or (iv) occurs, the determination of
L(1, χ) being quite complicated in these cases.
Let us first suppose that the character of signature v(λ) associated with χ
is given by v(λ) = 1 for all λ , 0 in K. In other words, for an integer α in
K, coprime to f, χ((α)) = χ(α). We may, moreover, suppose that f , (1),
since otherwise, χ is a character of the wide class group and L(s, χ) is just the
corresponding Dedekind zeta function of K.
Let z = x + iy be a complex variable with y > 0. Corresponding to an ideal
class A of K in the wide sense and an integer λ coprime to f, we define for
σ > 1, the function
X′
e2πi(muA +nvA ) |m + nz|−2s
g(z, s, λ, uA , vA ) = y s
m,n
where the summation is over all pairs of rational integers m, n not simultaneously zero. As a function of z, g(z, s, λ, uA , vA ) is not regular but as a function
of s, it is clearly regular for σ > 1. In fact, since uA and vA are not both integral, we know that g(z, s, λ, uA , vA ) has an analytic continuation which is an
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Applications to Algebraic Number Theory
104
entire function of s. For the sake of brevity, we shall denote it by gA (z, λ); this
does not mean that it is a function of z and λ depending only on the class A. It
does depend on the choice of the ideal bA , the integral basis [β1 , β2 ], the residue
class of λ modulo f and further on γ. But we have taken bA , β1 , β2 and γ to be
fixed.
In order to find L(1, χ), we shall employ the idea of Hecke already referred
P
e2πiS (λβγ) |N(β)|−s as
to on p. 86 (§ 3, Chapter II) and express the series
116
bA |(β),(0)
an integral over a suitable path in H, with the integrand involving gA (z, λ).
We shall denote the group of all units in K by Γ and the subgroup of units
congruent to 1 modulo f, by Γf . Moreover, let Γ∗f be the subgroup of Γf consisting of ρ such that ρ > 0. The group Γ∗f is infinite cyclic and let ǫ be the
generator of Γ∗f , which is greater than 1. We see that ǫ ≡ 1( mod f), N(ǫ) = 1,
ǫ > ǫ ′ > 0.
Since [ǫβ1 , ǫβ2 ] is again an integral basis of bA , ǫβ2 = aβ2 +bβ1 , ǫβ1 = cβ2 +
dβ1 where a, b, c, d are rational integers such that ad − bc = N(ǫ) = 1. Further,
if ω = β2 /β1 , then ǫω = aω + b, ǫ = cω + d so that ω = (aω + b)(cω + d)−1 .
We also have the same thing true for ω′ , namely, ǫω′ = aω′ + b, ǫ ′ = cω′ + d,
so that we have ω′ = (aω′ + b) · (cω′ + d)−1 . Thus the modular transformation
z → z∗ = (az+b)·(cz+d)−1 is hyperbolic, with ω, ω′ as fixed points and it leaves
fixed the semicircle on ω′ ω as diameter. If now we introduce the substitution
z = (ωpi + ω′ )(pi + 1)−1 (or equivalently, po = (z − ω′ )(ω − z)−1 we see that
when z(∈ H) lies on this semicircle, p is real and positive. As a matter of fact,
when z(∈ H) describes the semicircle form ω′ to ω, p runs over all positive real
numbers from 0 to ∞. Let p∗ i = (z∗ − ω′ ) · (ω − z∗ )−1 ; it is easy to verify that
p∗ = ǫ 2 p. This means that p∗ > p, since ǫ > 1. Consequently, whenever z
lies on the semicircle on ω′ ω as diameter, z∗ again lies on the semicircle and
always to the right of z.
Consider now
Z z∗
0
dp
F(A, λ, s) =
g(z, s, λ, uA , vA ) ,
p
z0
the integral being extended over the arc of this semicircle, from a fixed point
z0 ∈ H to z∗0 = (az0 + b)(cz0 + d)−1 . In view of the uniform convergence of the
P
series y s ′ m,n e2πi(muA +nvA ) |m + nz|−2s on this zrc we have, for σ > 1,
Z
z∗0
z0
d p X′ 2πiS (λβγ)
gA (z, λ)
e
=
P
m,n
Z
z∗0
z0
y s |m + nz|−2s
dp
,
P
where β = mβ1 + nβ2 . Setting µ = m + nω = β/β1 , we verify easily that
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Applications to Algebraic Number Theory
105
|m + nz|2 = (µ2 p2 + µ′ 2 )(p2 + 1)−1 and y = p(ω − ω′ )(p2 + 1)−1 . Thus
Z
z∗0
z0
y s |m + nz|−2s
dp
= (ω − ω′ ) s
p
Z
p∗0
P s (µ2 p2 + µ′ 2 )−s
p0
dp
p
Z
′ ∗
′ !
µ
ps
(ω − ω′ ) s |µ/µ |p0
dp
p → p
=
s
2
s
|N(µ)|
µ
|µ/µ′ |p0 (p + 1) p
√  s Z |(ββ′ /β′ β )|p ǫ 2

s
1
0
1
 N(bA ) D 
dp
p
= 
|
2 + 1) s p
′
′
|N(β)|
(p
|(ββ2 /β β1 )|p0
We have, therefore,
Z
X
√
e2πiS (λβγ) |N(β)|−s
F(A, λ, s) = (N(bA ) D) s
|(ββ′1 /β′ β1 )|ǫ 2 p0
ps
dp
s
+ 1) p
|(ββ′1 /β′ β1 )|p0
bA |β,0
(99)
We use once again the trick employed by Hecke (p. 86). For a fixed integer
β ∈ K, all the integers in K which are associated with β with respect to Γ∗f are
of the form βǫ ±k , k = 0, 1, 2, . . .. Keeping β fixed, if we replace β by βǫ l in the
integral on the right hand side of (99), then the integral goes over into
Z
|(ββ′1 /β′ β1 )|ǫ 2l+2 p0
p s (p2 + 1)−s
|(ββ′1 /β′ β1 )|ǫ 2l p0
(p2
dp
.
p
Now since ǫ > 1, the intervals (|ββ′1 |β′ β1 |p0 ǫ 2k , |ββ′1 |β′ β1 |p0 ǫ 2k+2 ) cover the
entire interval (0, ∞) without gaps and overlaps, as k runs over all rational
integers from −∞ to +∞. Moreover, since ǫ ≡ 1( mod f) and N(ǫ) = 1, we
k
have e2πiS (λβǫ γ) = e2πiS (λβγ) and |N(βǫ k )| = |N(β)|. Thus the total contribution
to the sum on the right hand side of (99) from all integers associated with a
fixed integer β with respect to Γ∗f , is given by
e2πiS (λβγ)
|N(β)| s
Z
0
∞
d p e2πiS (λβγ) Γ2 (s/2)
ps
=
|N(β)| s 2Γ(s)
(p2 + 1) s p
As a consequence,
118
√
(N(bA ) D) s Γ2 (s/2) X 2πiS (λβγ)
e
|N(β)|−s
F(A, λ, s) =
2Γ(s)
b |β(Γ∗ )
A
(100)
f
where, on the right hand side, the summation is over a complete set of integers
β , 0 in bA , which are not associated with respect to Γ∗f .
Applications to Algebraic Number Theory
106
Let {ρi } denote a complete set of units incongruent modulo f and let {ǫ j } be
a complete set of representatives of the cosets of Γf modulo Γ∗f . It is clear that
{ρi ǫ j } runs over a full set of representatives of the cosets of Γ modulo Γ∗f . Thus
F(A, λ, s) =
XX
√
Γ2 (s/2)
(N(bA ) D) s
e2πiS (λρi ǫ j βγ) |N(β)|−s
2Γ(s)
ρ ,ǫ b |(β)
i
j
(101)
A
where, on the right hand side, the inner sum is over all non-zero principal ideals
(β) divisible by bA .
By the signature of an element α , 0 in K, we mean the pair (α/|α|, α′ /|α′ |).
Now we can certainly find a set {µl } of integers µl such that µl ≡ 1( mod f)
and the set {ǫ j µl } consists precisely of 4 elements with the 4 possible different
signatures. Let {λk } be a set of integers λk constituting a complete system of
representatives of the cosets of G(f) modulo E(f). Here E(f) is the group of
prime residue classes mod f, containing at least one unit in K. The set {λk ρi }
is seen to be a complete set of representatives of the prime residue classes
modulo f. It is easy to prove
Proposition 15. The set of ideals {(λk µl )bA } serves as a full system of representatives of the ray classes modulo f.
Proof. Obviously it suffices to show that {(λk µl )} runs over a complete set of
representatives of the ray classes modulo f lying in the principal wide class. In
fact, if (α) ∈ Gf , then α ≡ ρi λk ( mod ∗ f) for some ρi and λk and moreover,
α
> 0. Since ǫ j µl ≡ 1( mod f), we have
we can find ǫ j µl such that
λk ρi ǫ j µl
(α) ∼ (λk µl ρi ǫ j ) = (λk µl modulo f. It is easy to verify that no two elements of
the set {(λk µl )} are equivalent modulo f.
From (101), we then have for σ > 1
X
χ(λk µl )χ(bA )F(A, λk µl , s)
λk ,µl ,A
=
√
Γ2 (s/2) X
·
χ(λk µl )χ(bA )(N(bA ) D) s ×
2Γ(s) λ ,µ ,A
k l
X
2πiS (λk µl ρi ǫ j βγ)
e
|N(β)|−s
×
bA |(β)
ρi ,ǫ j
=
√
Γ2 (s/2) X
χ(λk )χ(bA )(N(βA ) D) s ×
2Γ(s) λ ,ρ ,A
k i
X
X
×
χ(µl )
e2πiS (λk µl ρi ǫ j βγ) |N(β)|−s .
ǫ j ,µl
bA |(β)
(102)
119
Applications to Algebraic Number Theory
107
≡ 1( mod f), e2πiS (λk µl ρi ǫ j βγ) = e2πiS (λk ρi βγ) and since µl ≡ 1( mod f), χ(µl ) = 1.
Hence the sum in (102) is independent of ǫ j , µl and we have
X
χ(λk µl )χ(bA )F(A, λk µl , s)
λk ,µl ,A
Γ2 (s/2) s/2 X
χ(λk ρi )χ(bA )(N(bA )) s ×
D
2Γ(s)
λk ,ρi ,A
X
e2πiS (λk ρi βγ) |N(β)|−s .
=4·
bA |(β),(0)
Let (λk µl )bA lie in the ray class B modulo f. Denoting (λk µl )bA by bB ,
we know that bB runs over a full system of representatives of the ray classes
modulo f. Also, if α1 = λk µl β1 , α2 = λk µl β2 , then [α1 , α2 ] is an integral basis
of bB and we may now denote the function
X′
e2πiS ((mα1 +nα2 )γ) |m + nz|−2s
gA (z, λk µl ) = y s
m,n
R z∗
dp
by F(B, s).
p
The use of the notation F(B, s) is justified for, from (101), we see that
F(B, s) depends only on the ray class B modulo f (and on γ) and not on the
particular integral ideal bB chosen in B. For, if we replace bB by (µ)bB where
µ > 0, µ ≡ 1( mod ∗ f) and (µ)bB is an integral ideal coprime to f, then it is
easy to show that S (λk µl ρi ǫ j µβγ) − S (λk µl ρi ǫ j βγ) is a rational integer and in
addition, (N(bA )) s |N(β)|−s again depends only on the ideal class of bA .
Now χ((λk µl ))χ(bA ) = χ(bB ). Moreover, {λk ρi } runs over a complete set of
representatives λ of the prime residue classes modulo f. As a consequence, we
have
X
Γ2 (s/2)D s/2 X
χ(bB )F(B, s) = 2 ·
χ(λ)×
Γ(s)
B
λ mod f
X
X
χ(bA )(N(bA )) s
∗ ∗ ∗ ∗ ∗ ∗ (λβγ)|N(β)|−s
×
by gB (z, s) and
0
z0
gB (z, s)
A
bA |(β)
where B runs over all ray classes moudlo f ******** a complete system of
prime residue classes modulo f and A ***** ideal classes in the wide sense. In
other words, we have for σ **********,
X
χ(bB )F(B, s) = ∗ ∗ ∗ ∗ ∗(s/2)(Γ(s))−1 D s/2 T · L(s, χ).
(103)
B
120
Applications to Algebraic Number Theory
108
The function gB (z, s) is an Epstein zeta-function of the type of ζ(s, u, v, z, 0)
discussed earlier in § 5, Chapter I. Using the simple functional equation for the
Epstein zeta function ζ(s, u, v, z, 0), we shall now derive a functional equation
for L(s, χ).
Proposition 16. If χ is a proper ray class character modulo f(, (1)) whose
associated character of signature v(λ) ≡ 1, then L(s, χ) is an entire function of s satisfying the following functional equation, namely, if ξ(s, χ) =
π−s Γ2 (s/2)(DN(F)) s/2 L(s, χ), then
ξ(s, χ) =
χ(q)
ξ(1 − s, χ),
√
T/ N(f)
121
Remark. The functional equation (∗) for ****** generalization to arbitrary algebraic number **** derived by Hecke by using the generalized **** formula’
(for algebraic number ****** same, it is interesting to deduce the same ****
by using the functional equation satisfied by ζ(s, u, v, z, 0).
Proof. From (92), we have
π−s Γ2 (s/2)D s/2 T L(s, χ) = π−s Γ2 (s/2)D s/2
×
=
X
bA |(β),0
X
χ(λ)
λ
2πiS (λβγ)
e
X
χ(bA )(N(bA )) s
A
−s
|N(β)|
X
π−s Γ2 (s/2)D s/2 X
χ(λ)
χ(bA )(N(bA )) s ×
e(f)
A
λ
X
×
bA |β(Γ∗f )
A
where e(f) is the index of Γ∗f in Γ. Now, if we set u∗ = −v
uA , then, in the notation
∗
of § 5, Chapter I, we have gA (z, s, λ, uA , vA ) = ζ(s, u , 0, z, 0) and by (100)
X
2 X
χ(λ)
χ(bA )×
π−s Γ2 (s/2)D s/2 T L(s, χ) =
e(f) λ
A
Z z∗
0
dp
×
π−s Γ(s)ζ(s, u∗ , 0, z, 0) .
p
z0
By the method of analytic continuation of ζ(s, u∗ , 0, z, 0) discussed earlier in §
5, Chapter I, we have
Z z∗
0
dp
π−s Γ(s)ζ(s, u∗ , 0, z, 0)
p
Applications to Algebraic Number Theory
=
Z
z∗0
z0
dp
p
Z
0
∞
109


X′
 dt
−πiy−1 |m+nz|2 +2πi(muA +nvA ) 

 t s .
e

 t
m,n
R∞
R1
R∞
The inner integral 0 . . . may be split up as 0 · · · + 1 . . .. Now we may use
P −πy−1 t|m+nz|2 +2πi(muA +nvA )
e
and
the theta-transformation formula for the series
m,n
further make use of the inequality y−1 |m + nz|2 ≥ c(m2 + n2 ) uniformly on the
arc from z0 to z∗0 , for a constant c independent of m and n. If, in addition, we
keep in mind that χ is not the principal character, we can show that L(s, χ) is
an entire function of s.
Using now the functional equation of ζ(s, u∗ , 0, z, 0), we have
π−s Γ2 (s/2)D s/2 T L(s, χ) =
X
2 X
χ(λ)
χ(bA )×
e(f) λ
A
Z z∗
0
dp
×
π−(1−s) Γ(1 − s)ζ(1 − s, 0, u∗ , z, 0) .
p
z0
For σ < 0, ζ(1 − s, 0, u∗ , z, 0) = y1−s
P
m,n
|m − vA + (n + uA )z|−2(1−s) and by the
same arguments as above, we can show that
= Γ2
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗
!
X
√
1−s
|N(µ)|−(1−s)
(N(bA ) D)1−s
2
µ=β+ν
β∈b′A
√
where, on the right hand side, µ = −β′1 S (λβ2 γ) + β′2 S (λβ1 γ) = − DλγN(bA )
and µ runs over a complete set of numbers of the form β + ν with β ∈ b′A , such
that they are not mutually associated with respect to Γ∗f .
Let δ be an integer in K such that af = (δ) with an integral ideal a coprime
to f. Then, for σ < 0, we obtain from above that
π−s Γ2 (s/2)D s/2 T L(s, χ) =
π−(1−s) (1−s)/2 2
D
Γ ((1 − s)/2)×
e(f)
X
X
χ(bA )(N(bA ))1−s (N(af))1−s
χ(λ)×
×
A
×
X
β≡νδ( mod ab′A f)
β,0,ab′A |β(Γ∗f )
λ
−(1−s)
|N(β)|
,
122
Applications to Algebraic Number Theory
110
where, in the inner sum on the right hand side, β runs over a complete set of
integers in ab′A which are not mutually associated with respect to Γ∗f and which
satisfy β ≡ νδ( mod ab′A f). If β ≡ νδ mod ab′A f, then
√
√
χ(β) = χ(νδ) = χ(−γ DλδN(bA )) = χ((N(bA )γ Dλδ))
√
= χ(bA )χ(b′A )χ((γ Dδ))χ(λ) = χ(bA )χ(b′A )χ(aq)χ(λ).
123
Hence χ(λ) = χ(β)χ(bA b′A aq). Further if λ runs over representatives of prime
residue classes modulo f and β runs independently
over a complete set of inte√
gers in ab′A congruent modulo ab′A f to − DλγδN(bA ) and not mutually associated with respect to Γ∗f , then β runs over a complete set of integers in ab′A not
mutually associated with respect to Γ∗f and satisfying ((β), ab′A f) = ab′A , If we
note, in addition, that χ(β) = 0 for those β in ab′A for which (β)(ab′A )−1 is not
coprime to f, we can show finally that
X
X
χ(λ)
|N(β)|−(1−s) = χ(bA b′A aq)×
ab′A |β(Γ∗1 )
β≡νδ( mod ab′A F)
×
X
χ(β)|N(β)|−(1−s) .
ab′A |β(Γ∗f )
β,0
Thus, for σ < 0,
π−s Γ2 (s/2)D s/2 T L(s, χ) =
π−(1−s) (1−s)/2 2
D
Γ ((1 − s)/2)χ(q)×
(N(f)) s−1
X
×
χ(ab′A )N((ab′A ))1−s ×
A
X
χ(β)|N(β)|−(1−s)
ab′A |(β),(0)
= π−(1−s) D(1−s)/2 Γ2 ((1 − s)/2)
χ(q)
L(1 − s, χ),
(N(f)) s−1
since ab′A again runs over a system of representatives of the ideal classes similar to bA . Since L(s, χ) is an entire function of s, the above functional relation is 124
clearly valid for all s. We have now only to set ξ(s, χ) = π−s Γ s (s/2)(DN(f)) s/2 L(s, χ),
to see that the equation (∗) is true.
Let now [α1 , α2 ] be an integral basis for the ideal bB in the ray class B
and let ω = α2 /α1 , uB = S (α1 γ), vB = S (α2 γ). By Kronecker’s second limit
formula, we have
gB (z, s) = −π log |ϕ(vB , uB , z)|2
Applications to Algebraic Number Theory
111
+ · · · terms involving higher powers of (s − 1) . . . .
Letting s tend to 1, we have from (103),
L(1, χ) = −
2T
1
√
D
X
B
χ(bB )
Z
z∗0
z0
log |ϕ(vB , uB , z)|2
dp
.
p
Now it is easy to verify that
√
(ω − ω′ )
D
dp
=−
dz = −
dz,
p
(z − ω)(z − ω′ )
F B (z)
where
√
D
(z − ω)(z − ω′ ) = a1 z2 + b1 z + c1
(∗)
ω − ω′
has the property that a1 , b1 , c1 are rational integers with the greatest common
divisor 1, a1 > 0 and b21 − 4a1 c1 = D. Thus we have
√
Theorem 11. For a proper ray class character modulo f , (1) in Q( D),
D > 0, with the associated v(λ) ≡ 1, the value of L(s, χ) at s = 1 is given by
Z z∗
0
1 X
dz
,
χ(bB )
L(1, χ) =
log |ϕ(vB , uB , z)|2
2T B
F
B (z)
z0
F B (z) =
where B runs over all the ray classes modulo f and F B (z) is given by (∗) above.
As remarked earlier, the terms of the sum on the right hand side depend
only on γ (fixed!) and on the ray class B and not on the choice of the ideal bB
in B. We observe further that it does not seem to be possible to simplify this
formula any further, in an elementary way. One might try to follow the method
of Herglotz to deal with the integrals but then the invariance properties of the
integrand are lost in the process.
We proceed to discuss the case when the character of signature v(λ) associated with the given ray class character χ is of type (ii), i.e. for integral α(, 0)
N(α)
. In this case, we shall now see
coprime to f, we have χ((α)) = χ(α) ·
|N(α)|
that the value at s = 1 of L(s, χ) can be determined in terms of elementary
functions, using the first or second limit formula of Kronecker, according as
f = (1) or f , (1).
The fact that the character of signature v(λ) associated with χ is defined by
v(λ) =
N(λ)
for λ , 0
|N(λ)|
(104)
125
Applications to Algebraic Number Theory
112
implies automatically a condition on f. For instance, if ρ is a unit congruent to
N(ρ)
1 modulo, f, then
= χ(ρ)v(d) = χ((ρ)) = 1. In other words, every unit
|N(ρ)|
congruent to 1 modulo f should necessarily have norm 1.
To determine L(1, χ), we start as before from formula (92), namely, for
σ > 1,
X
X
χ(λ)
χ(bA )(N(bA )) s ×
L(s, χ) = T −1
λ
×
mod f
X
bA |(β),(0)
A
e2πiS (λβγ)
N(β)
|N(β)|−s .
|N(β)|
We shall try to express the inner sum on the right hand side as an integral, as
before. We shall follow the same notation as in the case treated above.
∂
1 ∂
i ∂
Let
=
−
; then in view of the uniform convergence of the
∂z
2 ∂x 2 ∂y
series defining the function gA (z, s, λ, uA , vA ) (denoted by gA (z, λ) for brevity)
for σ > 1, we have
∂gA (z, λ) X′ 2πi(muA +nvA ) ∂ s
e
=
{y |m + nz|−2s }
∂z
∂
z
m,n
X′
s
= y s−1
e2πi(muA +nvA ) |m + nz|−(2s−2) (m + nz)−2 .
2i
m,n
(105)
R z∗ ∂gA (z, λ)
dz, extended over the same
Let us now consider the integral z 0
0
∂z
arc of the semi-circle in H as considered earlier. Due to the uniform convergence of the series (105) on this arc, we have
Z
z∗0
z0
Z ∗
s X′ 2πi(muA +nvA ) z0
y s−1
∂gA (z, λ)
e
dz =
dz
2s−2 (m + nz)2
∂z
2i m,n
z0 |m + nz|
s
= D s/2 (N(bA )) s ×
2
Z p∗
X
0
p s−1 d p
×
e2πiS (λβγ) |N(β1 )|−s
.
2 2
′ 2 s−1 (µpi + µ′ )2
p0 (µ p + µ )
b |β,0
A
Effecting the substitution p → |µ′ /µ|p, we see that
Z
p∗0
p0
p s−1
dp
(µ2 p2 + µ′ 2 ) s−1 (µpi + µ′ )2
126
Applications to Algebraic Number Theory
−s
= |N(µ)|
where
τ=
Z
|µ/µ′ |p∗0
|µ/µ′ |p0
113
p s−1
d p,
(p2 + 1) s−1 (piτ + 1)2
N(µ)
N(β) N(β1 )
µ/µ′
=
=
= v(β)v(β1 ).
|µ/µ′ | |N(µ)| |N(β)| |N(β1 )|
Further (πτ + 1)2 = (pi + τ)2 . We have, therefore,
Z z∗
0 ∂g (z, λ)
s
A
dz = D s/2 (N(bZ )) s ×
∂z
2
z0
Z |(ββ′ /β′ β1 )|p0 ǫ 2
X
1
2πiS (λβγ)
−s
×
e
|N(β)|
|(ββ′1 /β′ β1 )|p0
bA |β,0
p s−1
d p.
(p2 + 1) s−1 (pi + τ)2
Now, since v(ǫ) = 1, the value of τ corresponding to β and βǫ k (k = ±1, ±2, . . .)
k
is the same. Moreover e2πiS (λβγ) = e2πiS (λβǫ γ) . If then we apply the same
analysis as in the former case, we obtain
Z z∗
0 ∂g (z, λ)
s
A
dz = D s/2 (N(bA )) s ×
∂z
2
z0
Z ∞
X
p s−1 d p
(106)
×
e2πiS (λβγ) |N(β)|−s
2
s−1 (pi + τ)2
0 (p + 1)
b |β(Γ∗ )
A
f
where, on the right hand side, the summation is over a complete set of integers
β , 0 in bA , which are not associated with respect to Γ∗f .
Effecting the transformation p → p−1 , we see that
Z ∞
Z ∞
p s−1
p s−1
d
p
=
−
d p.
2
s−1 (τ + ip)2
2
s−1 (τ − ip)2
0 (p + 1)
0 (p + 1)
Taking the arithmetic mean, we have
Z ∞
p s−1
dp
2
s−1 (τ + ip)2
0 (p + 1)
(
)
Z
1 ∞
p s−1
1
1
=
dp
−
2 0 (p2 + 1) s−1 (τ + ip)2 (τ − ip)2
Z ∞
dp
p s+1
= −2τi
2
s+1 p
(p
+
1)
0
v(β)v(β1 ) Γ2 ((s + 1)/2)
.
=
i
Γ(s + 1)
(107)
127
Applications to Algebraic Number Theory
114
From (106) and (107), we obtain for σ > 1,
Z z∗
0 ∂g (z, λ)
D s/2 Γ2 ((s + 1)/2)
A
v(β1 )
dz =
(N(bA )) s ×
∂z
2i
Γ(s)
z0
X
×
v(β)e2πiS (λβγ) |N(β)|−s .
bA |β(Γ∗f )
Since {ρi ǫ j } is a system of representatives of Γ modulo Γ∗f , we have
v(β1 )
Z
z∗0
z0
∂gA (z, λ)
D s/2 Γ2 ((s + 1)/2)
dz =
(N(bA )) s ×
∂z
2i
Γ(s)
X
v(βρi ǫ j )e2πiS (λβρi ǫ j γ) |N(β)|−s ,
×
128
(108)
bA |(β)
ρi ,ǫ j
where the summation on the right hand side is extended over all non-zero principal ideals divisible by bA and over the finite sets of representatives {ρi } and
{ǫ j }. Summing over all ideal classes A and the elements of the sets {λk } and {µl }
we obtain from (108),
Z z∗
X
0 ∂g (z, λ µ )
A
k l
χ(λk µl )χ(bA )v(β1 )
dz
∂z
z
0
λ ,µ ,A
k
=
l
D s/2 Γ2 ((s + 1)/2) X
χ(λk µl )χ(bA )(N(bA )) s ×
2i
Γ(s)
λk ,µl ,A
X
2πiS (λk µ1 βρi ǫ j γ)
v(βρi ǫ j )e
|N(β)|−s .
×
(109)
bA |(β)
ρi ,ǫ j
It is quite easy to verify that
v(ρi ǫ j ) = χ((ρi ǫ j ))χ(ρi ǫ j ) = χ(ρi ǫ j ) = χ(ρi ),
2πiS (λk µl βρi ǫ j γ)
e
2πiS (λk ρi βγ)
=e
.
Using these facts and applying the same arguments as in the earlier situation,
we see that the right hand side of (109) is precisely
X
Γ2 ((s + 1)/2) X
χ(λ)
χ(bA )(N(bA )) s ×
Γ(s)
A
λ mod f
X
2πiS (λβγ)
−s
×
v(β)e
|N(β)|
− 2iD s/2
bA |(β)
Applications to Algebraic Number Theory
= −2iD s/2
115
Γ2 ((s + 1)/2)
T · L(s, χ).
Γ(s)
To simplify the expression on the left hand side of (109), we first observe
that χ(λk µ1 ) = χ((λk µl ))v(λk µl ). Moreover, if we denote, as before, the ideal
(λk µl )bA by bB where B is the ray class modulo f containing it, then, by Proposition 15, bB runs over a full system of representatives of ray classes modulo
f. We may assume, without risk of confusion, that [β1 , β2 ] is still an integral basis of bB and set uB = S (β1 γ), vB = S (β2 γ). Further we may denote
P
gA (z, s, λk µl , uA , vA ) = y s ′ e2πi(muB +nvB ) |m + nz|−2s by gB (z) = gB (z, s). The
129
m,n
function gB (z, s) depends not just on the ray class B but also on the choice of
the ideal bB in B and on the integral basis [β1 , β2 ] of bB . Since bB and [β1 , β2 ]
are fixed, the modular transformation z → (az + b)(cz + d)−1 is uniquely determined by ǫβ2 = aβ2 + bβ1 , ǫβ1 = cβ2 + dβ1 ; ω = β2 /β1 , ω′ = β′2 /β′1 are the two
fixed points of this transformation and z∗0 = (az0 + b)(cz0 + d)−1 . From (109),
we have now, for σ > 1,
Z z∗
X
0 ∂g (z)
Γ(s)
iD−s/2
B
dz.
(110)
L(s, χ) =
χ(b
)v(β
)
B
1
2
2T Γ ((s + 1)/2) B
∂z
z0
Let us denote −
for σ > 1,
R z∗ ∂gB (z)
1
0
dz by G(B, s). We see from (108) that,
v(β
)
1
z0
∂z
2π2 i
G(B, s) =
D s/2 Γ2 ((s + 1)/2)
(N(bB )) s ×
4π2 Γ(s)
X X
√
v(βρi )e2π −1S (βρi γ) |N(β)|−s .
×
ρi ,ǫ j bB |(β),(0)
Obviously, the right hand side remains unchanged, when bB is replaces by another integral ideal in B. Thus, G(B, s) clearly depends only on B (and on γ
and s) but not on the special choice of bB or its integral basis. Writing χ(B) for
χ(bB ), we see that (110) goes over into
L(s, χ) =
X
π2
Γ(s)
χ(B)G(B, s).
s/2
T D ((s + 1)/2) B
130
(111)
∂gB (z, s)
is essentially an Epstein zeta-function of the type ζ(s, u, v, Q, P)
∂z
R z∗ ∂gB (z, s)
dz
discussed in § 5, Chapter I and it can be shown as before that z 0
0
∂z
The function
Applications to Algebraic Number Theory
116
is an entire function of s, by using the method of analytic continuation of the
Epstein zeta-function. Thus formula (110) gives the analytic continuation of
L(s, χ) into the entire s-plane, as an entire function of s. Moreover, if we use
∂gB (z, s)
the functional equation satisfied by
, we can show as in Proposition
∂z
16 that L(s, χ) satisfies the following functional equation, viz. if we set
Λ(s, χ) = π−s Γ2 ((s + 1)/2)(DN(f)) s/2 L(s, χ),
then
√
χ(q)v(γ D)
Λ(1 − s, χ).
(112)
Λ(s, χ) =
√
T/ N(f)
Let us observe that the constant on the right hand side is of absolute value
1.
In order to determine the value of L(1, χ), we distinguish between the two
cases f , (1) and f = (1).
(i) Let first f , (1). Then the numbers uB and vB corresponding to a ray
class B modulo f are rational numbers which are not both integral. For
the function gB (z, s) we have the following power-series expansion at
s = 1, by Kronecker’s second limit formula, namely
gB (z, s) = −π log |ϕ(vB , uB , z)|2
· · · (terms involving higher powers of (s − 1)).
Applying ∂/∂z to gB (z, s) and noting that (∂/∂z) log ϕ(vB , uB , z) = 0, we
∂gB (z, s)
have for
the power-series expansion,
∂z
∂gB (z, s)
∂
= −π log ϕ(vB , uB , z)
∂z
∂z
+ · · · (terms involving higher powers of (s − 1)).
Now since ϕ(vB , uB , z) is regular and non-vanishing in H, we can choose
a fixed branch of log ϕ(vB , uB , z) in H and (∂/∂z) log ϕ(vB , uB , z) is just
(d/dz) log ϕ(vB , uB , z). The coefficients of the higher powers of (s − 1)
might involve z. Letting s tend to 1, we have
z∗
v(β1 ) log ϕ(vB , uB , z) z00
2πi
We shall denote G(B, 1) by G(B) and then we have, from (111)
π2 X
L(1, χ) = √
χ(B)G(B).
T D B
G(B, 1) =
131
Applications to Algebraic Number Theory
117
P
(ii) Let now f = (1). Then, for all B, we see that gB (z, s) = y s m,n |m + nz|−2s
and by Kronecker’s first limit formula, we have the following powerseries expansion for gB (z, s) at s = 1, namely
gB (z, s) −
π
= 2π(C − log 2) − π log(y|η(z)|4 ) + · · · .
s−1
The application of ∂/∂z does away with the terms π/(s − 1) and 2π(C −
log 2). Noting that
∂
1
1
log y =
=
∂z
2iy z − z
we have for
namely
and
∂
(η(z))2 = 0,
∂z
∂gB (z, s)
, the following power-series expansion at s = 1,
∂z
∂gB (z, s)
π
d
=−
− π log(η(z))2 +
∂z
z−z
dz
+ · · · terms involving higher powers of (s − 1).
We wish to replace 1/(z − z) if possible by a regular function of z; but we
shall see now that we can do this when z lies on the semi-circle on ω′ ω
as diameter. In fact, the equation to the semi-circle on ω′ ω as diameter
is
z−ω
z−ω
+
= 0, z ∈ h.
′
z−ω
z − ω′
ω + ω′
, q = ωω′ , when
This means that zz + p(z + z) + q = 0 with p = −
2
z lies on the semi-circle. But then
1
z+ p
=
z − z z2 + 2pz + q
1 d
=
log(z2 + 2pz + q)
2 dz
p
d
log (z − ω)(z − ω′ ).
=
dz
Hence, for z on the semi-circle, we have the power-series expansion
p
∂gB (z, s)
d
= −π log (z − ω)(z − ω′ )η2 (z)+
∂z
dz
+ · · · (higher powers of s − 1).
132
Applications to Algebraic Number Theory
118
Thus letting s tend to 1,
iz∗
p
v(β1 ) h
log (z − ω)(z − ω′ )η2 (z) 0
z0
2πi
√
for a fixed branch of log( (z − ω)(z − ω′ )η2 (z)). Denoting G(B, 1) by
G(B) again, we have from (111),
G(B, 1) =
π2 X
χ(B)G(B).
L(1, χ) = √
D B
We are thus led to
133
√
Theorem 12. For a proper ray class character χ(, 1) modulo f in Q( D)(D >
0), with associated v(λ) defined by (104), we have
L(1, χ) =
T
π2
√
D
X
χ(B)G(B)
B
where the summation is over all ray classes B modulo f and
 v(β ) z∗0

1



 2πi log ϕ(vB , uB , z) z0 , for f , (1),

G(B) = 

iz∗
√

v(β ) h


 1 log( (z − ω)(z − ω′ )η2 (z)) 0 , for f = (1).
z0
2πi
(113)
We are now interested in the explicit determination of the values of G(B)
corresponding to the various ray classes B, in terms of elementary arithmetical
functions. The expressions inside the square brackets in (113) represent analytic functions of z and we know that G(B) itself does not depend on the point
z0 chosen on the semi-circle on ω′ ω as diameter. Thus G(B) is, in both cases,
the value of an analytic function z, which is a constant on the semi-circle on
ω′ ω as diameter; as a consequence, G(B) is a constant independent of z0 and in
order to calculate G(B), we might replace z0 in (113) by any point z in H, not
necessarily lying on the semi-circle on ω′ ω as diameter.
∗
First we observe that (1/2πi)[log ϕ(vB , uB , z)]zz is a rational number with at
most 12 f in the denominator, where f is the smallest positive rational integer
divisible by f. This can be verified as follows. Let z∗ = (az + b)(cz + d)−1
where the rational integers a, b, c, d satisfying the condition ad − bc = 1 are
uniquely determined by the unit ǫ and a fixed integral basis [β1 , β2 ] of a fixed
integral ideal bB in B, coprime to f. Corresponding to z∗ , let v∗B = avB + buB
and u∗B = cvB + duB . We know already from the properties of ϕ(vB , uB , z) that
Applications to Algebraic Number Theory
119
ϕ(v∗V , u∗B , z∗ ) = ρϕ(vB , uB , z) where ρ is a 12th root of unity depending only on
a, b, c, d. On the other hand v∗B = aS (β2 γ) + bS (β1 γ) = S (ǫβ2 γ) and hence
v∗B − vB = S ((ǫ − 1)β2 γ) is a rational integer k, say; similarly, u∗B − uB is again
a rational integer l, say. But we know from (99) that
ϕ(v∗B , u∗B , z∗ ) = ϕ(vB + k, uB + l, z∗ )
= τϕ(vB , uB , z∗ ),
where τ is a 2 f -th root of unity. Hence
134
ϕ(vB , uB , z∗ ) = θϕ(vB , uB , z),
where θ is a 12 f -th root of unity. Choosing a fixed branch of log ϕ(vB , uB , z),
1
∗
we see that
[log ϕ(vB , uB , z)]zz is a rational number with at most 12 f in the
2πi
denominator. Now, we know that for z ∈ H, the function ϕ12 f (vB , uB , z) is
∗
a modular function of level f and [log ϕ(vB , uB , z)]zz is just a ‘period’ of the
abelian integral log ϕ(vB , uB , z). The explicit computation of these ‘periods’ of
the abelian integral log ϕ(vB , uB , z) has been essentially considered by Hecke,
in connection with the determination of the class number of a biquadratic field
obtained from a real quadratic field k over Q, by adjoining the square root
of a “totally negative” number in k; employing the ideal of the well-known
Riemann-Dedekind method, Hecke used for this purpose, the asymptotic behaviour of log ϑ11 (w, z) as z tends to infinity and z∗ to the corresponding ‘rational point’ a/c on the real axis. It is to be remarked, however, that from
Hecke’s considerations, one can at first sight conclude only that the quantities
1
∗
[log ϕ(vB , uB , z)]zz are rational numbers with at most 24 f 2 in the denomina2πi
tor.
√
1
∗
[log (z − ω)(z − ω′ )η2 (z)]zz , we can conclude again that
Regarding
2πi
they are rational numbers with at most 12 f in the denominator. For this purpose, we notice that
z∗ − ω =
z−ω
ǫ(cz + d)
and
z ∗ − ω′ =
z − ω′
+ d)
ǫ ′ (cz
and η2 (z∗ ) = ρ(cz + d)η2 (z), ρ being a 12th root of unity depending only on a,
b, c, d. Hence
p
p
(z∗ − ω)(z∗ − ω′ )η2 (z∗ ) = ρ (z − ω)(z − ω′ )η2 (z).
√
Choosing a fixed branch of log( (z − ω)(z − ω′ )η2 (z)) in H, we see that our
assertion is true.
Applications to Algebraic Number Theory
120
We may now proceed to obtain the value of G(B), first for f , (1).
We make a few preliminary simplifications. We can suppose without loss
of generality that 0 ≤ uB , vB < 1, since, in view of (98), ϕ(vB , uB , z) merely
picks up a 2 f th root of unity as a factor when vB is replaced by vB ± 1 or uB by
∗
uB ± 1 and this gets cancelled, when we consider [log ϕ(vB , uB , z)]zz . Moreover,
since we know already that G(B)v(β1 ) is real (in fact, even rational) it is enough
1
∗
to find the real part of
[log ϕ(vB , uB , z)]zz . Now
2πi
ϕ(v, u, z) = eπiu(uz−v)−πi/2+πiz/6 (eπi(v−uz) − e−πi(v−uz) )×
∞
∞
Y
Y
×
(1 − e2πi(v−uz)+2πimz )
(1 − e−2πi(v−uz)+2πimz )
m=1
m=1
= eπiu(uz−v)−πi/2+πiz/6+πi(v−uz) ×
∞
∞
Y
Y
×
(1 − V Qm−u )
(1 − V −1 Qm+u ),
m=1
m=0
where we have set V = e2πiv , Q = e2πiz and Q−u = e−2πiuz . Moreover, if u = 0,
then 0 < v < 1 and in the second infinite product above we notice that m+u ≥ 0,
in any case. We define the branch of (1/2πi) · log ϕ(v, u, z) by
!
z 2
1
1 v
1
log ϕ(v, u, z) =
u −u+
− + (1 − u)+
2πi
2
6
4 2
∞
1 X
log(1 − V Qm−u )+
+
2πi m=1
+
1 X
log(1 − V −1 Qm+u ),
2πi m=0
∞
where, on the right hand side, we take the principal branches of the logarithms.
∞ (V −1 Qu )n
P
Noting that the series
converges even if u = 0, since then 0 < v <
n
n=1
m−u n
∞ (V −1 Qm+u )n
∞ P
∞ P
P
P
)
∞ (V Q
and
converge
1 and further the series
n=1
n
n
m=1 n=1
m=1
absolutely, we conclude that
!
1
1
1 v
1 2
u −u+
z − + (1 − u)−
log ϕ(v, u, z) =
2πi
2
6
4 2
∞
−u n n
X
1 (V Q ) Q + (V −1 Qu )n
1
−
2πi n=1 n
1 − Qn
135
Applications to Algebraic Number Theory
121
Thus, for f , (1),
G(B)v(β1 ) =
136
!
1 ∗
1 2
u − uB +
(z − z)−
2 B
6
∞
∗
X 1 (V Q 12 −uB )n + (V −1 Q− 12 +uB )n z

− 
2πin
Q−n/2 − Qn/2
n=1
(114)
z
2πivB
where we have used V to stand for e
without risk of confusion.
Now c = (ǫ − ǫ ′ )/(ω − ω′ ) > 0; so, if we set z = −(d/c) + (i/cr) with r > 0,
then z∗ = (a/c) + (ir/c). As r tends to zero, z∗ tends to the rational point a/c
vertically and z tends to infinity. In view of our earlier remarks, we may let
r tend to zero in (114), in order to find the value of G(B)v(β1 ); moreover, it
suffices to find the real part of the right hand side of (114), since G(B)v(β1 ) is
rational. Thus, we have for f , (1),
!
1 (a + d)
1 2
+ σ1 + σ2
uB − uB +
G(B)v(β1 ) =
2
6
c
where


−u
u
∞

1 X 1 (V Q1 B )n Qn1 + (V −1 Q1B )n 

σ1 = lim real part of
 ,
r→0
2πi n=1 n
1 − Qn1


1
1
∞
−1 − 2 +uB n 
2 −u B

X

)
+
(V
Q
)
(V
Q
1
−1
2
2

σ2 = lim real part of
−n/2
n/2

r→0
2πi n=1 n
Q2 − Q2
∗
with Q1 = e2πiz and Q2 = e2πiz .
As r tends to zero, Q1 tends to zero exponentially and it is easy to see that
σ1 = 0 unless uB = 0 and in this case
σ1 =
∞
X
V −n − V n
4πin
n=1
So, if we define λ(uB ) to be zero if 0 < uB < 1 and equal to 1 if uB = 0, then
σ1 = λ(uB )
∞
X
V −n − V n
n=1
4πin
.
In order to evaluate σ2 , we need the identity
qc − q−c X c−2k+1 X −(c−2k+1)
=
q
=
q
,
q − q−1
k=1
k=1
c
c
(q , q−1 )
137
Applications to Algebraic Number Theory
i.e.
122
X q−(c−2k+1)
X qc−2k+1
1
=
.
=
q − q−1 k=1 qc − q−c k=1 qc − q−c
c
c
(115)
Employing the identities (115) with q = Q−n/2
, we have
2


k−c/2−uB n
∞
c X
B −n
X

) + (V Qk−c/2−u
) 
1 (V Q2
2

 .
σ2 = − lim real part of
r→0
2πin
Q−cn/2 − Q+cn/2
k=1 n=1
2
2
Let now tk = 1 − (2/c)(k − uB ) for k = 1, 2, . . . c; clearly −1 ≤ tk < 1 for
0 ≤ uB < 1. Further, let for k = 1, . . . , c, Vk = e2πi(vB +(a/c)(k−uB )) and let P = eπr .
It is clear that Qc2 = e2πia−2πr = P−2 and P tends to 1 as r tends to zero. Now it
can be verified that


∞
c X
X

1 (Vk Ptk )n + (Vk Ptk )−n 
σ2 = − lim real part of

P→1
2πin
Pn − P−n
k=1 n=1
i.e.
∞
c X
X
1 (Vkn − Vk−n )(Ptk n − P−tk n )
.
P→1
4πin
Pn − P− n
k=1 n=1
σ2 = − lim
(116)
Ptk n − P−tk n
= tk and if we can establish the uniform
P→1 Pn − P−n
convergence of the inner series in (116) with respect to P for 1 ≤ P < ∞,
then we can interchange the passage to the limit and the summations in (116).
For this purpose, we remark that, for fixed k with 1 ≤ k ≤ c, the function
xtk − x−tk
fk (x) =
is a bounded monotone function of x for 1 ≤ x < ∞. In fact,
x − x−1
it is monotone decreasing for 0 < tk < 1, identically zero for tk = 0, monotone
increasing for −1 < tk < 0 and identically ±1 for tk = ±1. Moreover if |tk | < 1,
fk (x) tends to 0 as x tends to infinity and fk (x) tends to tk as x tends to 1.
Thus the sequence { fk (Pn )}, n = 1, 2, . . . is a bounded monotone sequence for
∞ V n − V −n
P
k
k
is convergent. By Abel’s criterion,
1 ≤ k ≤ c. Further the series
4πin
n=1
the inner series in (116) converges uniformly with respect to P for 1 ≤ P < ∞.
As a consequence, we have from (116),
! ∞
c
X
k − uB 1 X Vkn − Vk−n
σ2 =
−
c
2 n=1 2πin
k=1
We know that lim
Let now for real x,
P1 (x) = −
∞
X
e2πinx − e−2πinx
n=1
2πin
;
138
Applications to Algebraic Number Theory
as is well-known, we have

1



 x − [x] −
P1 = 
2


0
123
for x not integral,
for x integral,
where [x] denotes the integral part of x. Further, let for real x,
P2 (x) = −
∞
X
′ e2πinx
.
(2πin)2
n=−∞
Clearly P2 (x) = 12 {(x − [x])2 − (x − [x])} + 1/12. With this notation then, we
have for f , (1),
(a + d) 1
+ λ(uB )P1 (vB )−
c
2
!
c
X k − uB 1 !
k − uB
P1 a ·
−
+ vB .
−
c
2
c
k=1
G(B)v(β1 ) = P2 (uB )
Now, for 1 ≤ k ≤ c and 0 ≤ uB ≤ 1, 0 < (k − uB )/c < 1 and (k − uB )/c = 1 only
when k = c and uB = 0. Therefore, for 1 ≤ k < c,
!
k − uB
k − uB 1
P1
=
−
c
c
2
and for k = c,
P1
139
!
k − uB 1 1
k − uB
=
− − λ(uB ).
c
c
2 2
Further when k = c and uB = 0,
!
k − uB
+ vB = P1 (a + vB ) = P1 (vB ).
P1 a ·
c
Thus
(a + d) 1
+ λ(uB )P1 (vB )−
c
2
!
!
c
X
k − uB
1
k − uB
P1 a ·
+ vB − λ(uB )P1 (vB )
−
P1
c
c
2
k=1
!
!
c
(a + d) X′
k − uB
k − uB
= P2 (uB )
P1 a ·
−
+ vB .
P1
c
c
c
k=1
G(B)v(β1 ) = P2 (uB )
(117)
Applications to Algebraic Number Theory
124
Since P1 (x) is an odd function of x,
!
!
k − uB
−k + uB
P1
= −P1
c
c
and
!
!
−k + uB
k − uB
+ vB = −P1 a ·
− vB .
P1 a ·
c
c
Moreover, in the sum in (117), we can lek k run over any complete set of
residues modulo c, in view of the periodicity of P1 (x). Thus
!
!
c
(a + d) X
−k + uB
−k + uB
G(B)v(β1 ) = P2 (uB )
P1 a ·
−
− vB
P1
c
c
c
k=1
!
!
c
k + uB
k + uB
(a + d) X
−
−v
P1 a ·
P1
= P2 (uB )
c
c
c
k=1
!
!
c−1
(a + d) X
k + uB
k + uB
= P2 (uB )
−
− vB . (118)
P1 a ·
P1
c
c
c
k=0
140
It is surprising that even though, prima facie, it appears from (118) that
G(B) is a rational number with a high denominator, say 12c2 f 2 , the factor c2
in the denominator drops out and eventually, G(B) is a rational number with at
most 12 f in the denominator.
The determination of the asymptotic development of log ϑ11 (vB − uB z, z) as
z tends to a/c is done in a rather more complicated manner by Hecke in his
work referred to earlier.
We now take up the calculation of G(B) in the case f = (1). We set z∗ =
a/c + ir/c and z = −(d/c) + i/cr with r > 0 and as before, G(B)v(β1 ) is
iz∗ !
p
1 h
log (z − ω)(z − ω′ )η2 (z) .
lim real part of
z
r→0
2πi
We shall show first that
iz∗ !
p
1
1 h
′
=− .
lim real part of
log (z − ω)(z − ω )
z
r→0
2πi
4
For this purpose, we remark in the first place that −(d/c) < ω′ < ω < a/c,
since ω(a − cω) = 1 implies a/c > ω and ǫ(cω′ + d) = 1 implies −d/c < ω′ .
Now as z = −(d/c) + i/cr and r tends to zero, arg (z − ω)(z − ω′ ) tends to π and
Applications to Algebraic Number Theory
125
similarly arg(z∗ − ω)(z∗ − ω′ ) tends to zero as z∗ = a/c + ir and r tends to zero.
Thus our assertion above is proved.
Defining the branch of log η2 (z) by
log η2 (z) =
∞
X
πiz
+2
log(1 − e2πimz ),
6
m=1
where on the right hand side we have taken the principal branches, we see that
log η2 (z) =
∞
∞ X
X
1 2πimnz
πiz
−2
e
6
n
m=1 n=1
and in view of absolute convergence of the series, we have once again,
log η2 (z) =
∞
X
πiz
1 Qn
−2
,
6
n 1 − Qn
n=1
where Q = e2πiz . Now
∗
∞
h
iz∗ πi
X 1 Qn z
∗
2

log η (Z) = (z − z) − 2 
z
6
n 1 − Qn 
n=1
z
By our remarks above,
1
1 (a + d)
G(B)v(β1 ) = − +
− σ3 ,
4 12 c
where


σ3 = 2 · lim real part of
r→0
It is easily seen that
 ∗
∞
n z 
X 1
Q
 

  .
n 
2πin
1
−
Q
n=1
∞
X
Qn1
1
= 0,
r→0
2πin 1 − Qn1
n=1
lim
z
for
Q1 = e2πiz .
Thus


∞
X
Qn2 
1
1

 with Q2 = e2πiz∗ .
σ3 = lim real part of
n
r→0
2
2πin 1 − Q2
n=1
141
Applications to Algebraic Number Theory
126
To determine σ3 , we first use the identity,
X qc−2k+1
1
,
=
q − q−1 k=1 qc − q−c
c
to make the denominators of the terms Qn2 /(1 − Qn2 ) real. Setting q = Q−n/2
in
2
this identity, we see that
−n/2(c−2k)
∞
r
∞
X
X
Qn2
1
1 X Q2
=
.
2πin 1 − Qn2 n=1 2πin k=1 Q−nc/2 − Qnc/2
n=1
2
2
Now Q−nc/2
= (−1)an Pn with P = eπr and Qk2 = Vk P−2k/c where Vk = e2πi(a/c)k .
2
Hence
∞
∞
c X
X
X
Qn2
1
1 Vkn Pn(1−2k/c)
.
=
2πin 1 − Qn2 k=1 n=1 2πin Pn − P−n
n=1
As a consequence,
 c ∞

X X 1 (Vkn − Vk−n )Ptk n 

 ,
σ3 = lim 
r→0
2πin Pn − P−n 
k=1 n=1
(119)
1 (Vkn − Vk−n )Ptk n
corresponding
Pn − P−n
n=1 2πin
to k = c is zero since Vcn − Vc−n = 0. Hence effectively k runs from 1 to c − 1;
but then c − k also does the same when k does so. On the other hand, if k is
replaced by c − k, then Vk goes to Vk−1 and tk to −tk so that
where tk = 1 − 2k/c. In (119), the sum
∞
P
∞
c X
∞
c X
X
X
1 (Vkn − Vk−n )Ptk n
1 (Vkn − Vk−n )P−tk n
=
−
2πin Pn − P−n
2πin
Pn − P−n
k=1 n=1
k=1 n=1
Taking the arithmetic mean, we see that
σ3 =
∞
c X
X
1
1 (Vkn − Vk−n )(Ptk n − P−tk n )
lim
2 r→0 k=1 n=1 2πin
Pn − P−n
We are now in the same situation as in (116) but with uB and vB replaced by 0.
By the same analysis as in the former case, we can show that
σ3 =
c
X
k=1
P1
!
! X
!
!
c−1
k
k
k
k
P1 a ·
=
P1 a · .
P1
c
c
c
c
k=0
142
Applications to Algebraic Number Theory
127
Thus, for f = (1),
!
!
c−1
1
k
k
(a + d) X
G(B)v(β1 ) = − + P2 (0)
P1
P1 a · .
−
4
c
c
c
k=0
We now define

1



 ,
v(f) = 
4


0,
(120)
143
if f = (1)
otherwise.
Consolidating (118) and (120) into a single formula, we see that the value of
G(B) is given explicitly in terms of elementary arithmetical functions by
Theorem
13. Corresponding to a proper ray class character χ modulo f in
√
Q( D)(D > 0), with associated v(λ) defined by (104) and a ray class B modulo
f, we have the formula


!
!
c−1




k + uB
k + uB
a+d X


,
−
− vB − v(f)
P1 a
P1
G(B) = v(β1 ) 
P2 (uB )




c
c
c
k=0
(121)
where uB = vB = 0 for f = (1).
Note . Here [β1 , β2 ] is an integral basis of an integral ideal bB in B and uB =
S (β1 γ), vB = S (β2 γ). We recall that G(B) depends only on B and not on the
special choice of bB or of its integral basis. The rational integers a, b, c, d are
determined by ǫβ2 = aβ2 + bβ1 , ǫβ1 = cβ2 + dβ1 . The Riemann-Dedekind
method used above for the explicit determination of G(B) does not make any
specific use of the fact that z → z∗ is a hyperbolic substitution.
Coming back
√ to our formula for L(1, χ) for a proper ray class character χ
modulo f in Q( D) for which the associated v(λ) is defined by (104), we have
as a consequence of Theorems 12 and 13,
L(1, χ) =
where Λ =
P
π2
√ Λ,
T D
(122)
χ(B)G(B) with B running over all ray classes modulo f. G(B) is
B
given by (121), and
T=
λ
√
X
χ(λ)e2πiS (λγ)
mod f
with γ ∈ K such that (γ D) has exact denominator f.
144
Applications to Algebraic Number Theory
128
We shall now use the elementary formula for L(1, χ) derived above, in order
to determine
√ the class number of special abelian extensions of the real quadratic
field Q( D).
√
Let F be a relative abelian extension of degree n over K = Q( D) and
let f be the ‘Führer’ (conductor) of F relative to K. Let ζ(s, K) and ζ(s, F) be
the Dedekind zeta functions of K and F respectively. From class field theory,
we know that the galois group of F over K is isomorphic to a subgroup of the
character group of the group of ray classes modulo f in K. Moreover, the law of
splitting in F of prime ideals of K is equivalent to the following decomposition
of ζ(s, F), namely
Y
ζ(s, F) = ζ(s, K)
L(s, χ)
(123)
χ,1
where χ runs over a complete set of n − 1 non-principal ray class characters
modulo fχ with conductor fχ dividing f and L(s, χ) is the L-series in K associated with χ. Multiplying both sides of (123) by s − 1 and letting s tend to 1,
we have in analogy with Theorem 10,
2r1 · (2π)r2 · R · H
4rh Y
= √
L(1, χ).
√
W |∆|
w D χ,1
(124)
In (124), r1 and 2r2 are respectively the number of real and complex conjugates
of F over Q, H is the class number of F, W is the number of roots of unity in F,
R is the regulator of F, ∆ is the discriminant of F over Q, h is the class number
of K, r is the regulator of K and w(= 2) the number of roots of unity in K.
We have been able to obtain a formula for L(1, χ) involving elementary
arithmetical functions only in the case when the character of signature v(λ) associated with χ is defined by (104). Thus, if we are to obtain for H, a formula
involving purely elementary arithmetical functions, then we ought to consider
only such abelian extensions F over K, for which, on the right hand side of
(124), there occur in the product only characters χ whose associated v(λ) is
given by (104). Moreover, suppose that χ1 and χ2 are two such distinct characters occuring in that product, then χ1 χ−1
2 also occurs in the product and its
associated v(λ) is defined by v(λ) = 1 for all λ , 0 in K. But this, again, is
a situation which we should avoid, in view of our aim. Thus, in order that we
could obtain a formula of elementary type for the class number H of F, we
are obliged to consider only those abelian extensions F over K, for which we
have the decomposition ζ(s, F) = ζ(s, K)L(s, χ) where χ is a ray class character modulo fχ with fχ as conductor and its associated character of signature is
given by (104). This,
√ in
√ the first place, means that F is a quadratic extension
of K, i.e. F = Q( D, θ) where θ is a non-square number in K. Let ϑ be the
145
Applications to Algebraic Number Theory
129
relative discriminant of F over K. We could assume that (θ) = ϑi2 for an ideal
i in K, coprime to ϑ.
From class field theory, we know that a prime ideal p in K not dividing f,
the conductor of F relative to K, either splits into a product of distinct prime
factors or stays prime in F and moreover



if p splits in F
1,
(125)
χ(p) = 

−1, if p stays prime in F.
We also know that fχ divides f.
On the other hand, let p be a prime ideal in K not dividing ϑ. We can then
find c ∈ K such that θ∗ = θc2 is prime to p. Let us define then
!
θ∗
ψ(p) =
,
p
!
θ∗
where, for p 6 |(2),
is the quadratic residue symbol in K and if pa is the
p
!
θ∗
= +1 or −1, according as the conhighest power of p dividing (2), then
p
gruence θ∗ ≡ ξ2 ( mod p2a+1 ) is solvable in K or not. We see that ψ(p) is
unambiguously defined for all prime ideals p not dividing ϑ and we extend ψ
multiplicatively to all ideals a in the ray classes modulo ϑ in K.
From the theory of relative quadratic extensions of algebraic number fields,
it follows that for prime ideals p in K not dividing ϑ, χ(p) = ψ(p) and hence
χ(a) and ψ(a) coincide on the ideals in the ray classes modulo ϑ.
Now, by the law of quadratic reciprocity in K, it can be shown that for
all numbers α in K for which α ≡ 1( mod ∗ ϑ) and α > 0, ψ((α)) = 1. In
other words, ψ, and hence χ, is a ray class character modulo ϑ. Incidentally
we note that fχ divides ϑ, since χ is a ray class character modulo fχ , with fχ as
conductor. Again, using the law of quadratic reciprocity, we can show that for
integral λ , 0 and coprime to ϑ, χ((λ)) = ψ((λ)) = ψ(λ)v(λ) where ψ(λ) is a
prime residue class character modulo ϑ and v(λ) is given by


1, if θ > 0








λ



, if θ > 0, θ′ < 0



 |λ|
v(λ) = 

λ′



, if θ < 0, θ′ > 0

′|


|λ





N(λ)


, if − θ > 0.

|N(λ)|
146
Applications to Algebraic Number Theory
130
N(λ)
. Thus,
|N(λ)|
finally, in order that we could calculate H in terms of elementary
arithmetical
√
functions, we conclude that F should be of the form K( θ) where θ ∈ K and
−θ > 0. In other words, F is a biquadratic field over Q, which√is realized as an
imaginary quadratic extension of the real quadratic field Q( D). We have in
this case, the following formula for the class number H of F, viz.
But, for our purposes, we require χ((λ)) to be of the form ψ(λ) ·
4π2 · R
4rh
√ H = √ L(1, χ)
W |∆|
w D
4rh π2
= √
√ Λ.
w DT D
(126)
In order to determine the value of T explicitly,
we use the functional equa√
tions of ζ(s, F), L(s, χ) and ζD (s) = ζ(s, Q( D)), viz.
(2π)−2s Γ2 (s)(|∆|) s/2 ζ(s, F) = (2π)−2(1−s) Γ2 (1 − s) × (|∆|)(1−s)/2 ζ(1 − s, F),
!
1 − s (1−s)/2
D
ζD (1 − s)
(127)
π−s Γ2 (s/2)D s/2 ζD (s) = π−(1−s) Γ2
2
π−s Γ2 ((s + 1)/2)(DN(fχ )) s/2 L(s, χ) =
!
−(1−s) 2 1 − s
(DN(fχ ))(1−s)/2 L(1 − s, χ)×
=π
Γ
2
√ p
χ(q)v(γ D) N(fχ )
,
T
√
where γ ∈ K such that (γ D) = q/fχ has exact denominator fχ . Further
1
ζ(s, F) = ζD (s)L(s, χ) and Γ(s/2)Γ((s + 1)/2) = π 2 21−s Γ(s). These, together
with (127), give us
√ p
! s− 1
D2 N(fχ ) 2 χ(q)v(γ D) N(fχ )
.
(128)
=
|∆|
T
Sincepthe right hand√side is independent of s, we see by setting s = 1/2, that
T = N(fχ )χ(q)v(γ d). Since the right-hand side of (128) is 1, we see again
by setting s = 3/2 in (128), that D2 N(fχ ) = |∆|. But we know that D2 N(ϑ) =
|∆|. This means that N(ϑ) = N(fχ ) and since fχ divides ϑ, we obtain incidentally
that fχ = ϑ. Thus we have
p
√
T = N(ϑ)χ(q)v(γ D).
147
Applications to Algebraic Number Theory
But
131
√
q
qi2
(κ)
q
= = 2 =
(γ D) =
fχ ϑ ϑi
(θ)
√
2
where κ = θγ D. Moreover, since√ χ is a real
√ character, χ(i ) = √1. Hence
2
χ(q) = χ(qi ) = χ(κ)v(κ) = χ(κ)v(θγ D) = v(γ D)χ(κ). Thus T = N(ϑ)χ(κ).
As a result, from (126), we obtain
HRw
= χ(κ)Λ
h rW
√ X
χ(B)G(B).
= χ(θγ D)
(129)
B
We may summarise the above in
√ √
Proposition
√ 17. Let F = Q( D, θ) be an imaginary quadratic extension of
K = Q( D), D > 0, with an integral ideal f in K for its “conductor”, θ being a
totally negative number in K. Then with our earlier notation, the class number
H of F is given by
H=h
√ X
rW
χ(B)G(B),
χ(θγ D)
Rw
B
where B runs over all the ray classes modulo f in K, G(B) is given by (121) and
χ is the non-principal ray class character modulo f, associated with F.
Remark. We know that G(B) are rational numbers with at most 12 f (where f
is the smallest positive rational integer divisible by fχ = f = ϑ) in the denomiR w H
is a rational integer.
nator, it follows that 12 f
r W h
√
Let η and µ be respectively the fundamental units in F and Q( D). Then
R = 2 log |η| and r = log |µ|. Now either µ = η or µ = ±η√2 . In the former case
R = 2r; in the latter case, F may be generated over Q( √ D) by adjoining the
square root of ∓µ (whichever is totally negative) to Q( D) and in this case,
we have R = r. Thus, in any case, since w divides W, we see that 24 f (H/h) is
a rational integer.
If h is coprime to 24 f , we obtain the interesting result that h divides H.
Let√us now assume that F is an unramified imaginary quadratic extension
of Q( D). Then ϑ = (1) and
√ hence the ray class group modulo ϑ is just
the narrow class group of Q( D) and χ is a genus character.√ It is known
that all totally
complex unramified quadratic extensions of Q( D) are of the
√
√
form Q( D1 , D2 ) where D1 and D2 are coprime negative discriminants (of
148
Applications to Algebraic Number Theory
132
quadratic
fields over Q) which
√
√ satisfy√D1 D2 = D. Hence F is obtained from
Q( D) by adjoining either D1 or D2 , for such a decomposition of D as
product of two coprime negative discriminants.
√
√
On the other hand, since F = Q( D1 , D2 ) is an abelian extension of Q
with galois group of order 4, we have the decomposition
ζ(s, F) = ζ(s)LDt (s)LD2 (s)LD (s),
where LD1 (s), LD2 (s) and LD (s) are the Dirichlet L-series in Q associated with
D D D
1
2
the Legendre-Jacobi-Kronecker symbols
,
, and
respectively.
n
n
n
Moreover
ζD (s) = ζ(s)LD (s)
and hence
149
L(s, χ) = LD1 (s)LD2 (s),
a result due to Kronecker,
√ which we have met already (§1, Chapter II), χ being
a genus character in Q( D). Therefore
L(1, χ) = LD1 (1)LD2 (1).
√
√
On the other hand, if h1 and h2 are the class numbers of Q( D1 ) and Q( √D2 )
respectively
√ and w1 and w2 the respective number of roots of unity in Q( D1 )
and Q( D2 ), then
LD1 (1) =
2πh1
,
√
w1 |D1 |
LD2 (1) =
2πh2
.
√
w2 |D2 |
Hence
4π2 h1 h2
π2
√ Λ = L(1, χ) =
√
√
w1 w2 |D1 | |D2 |
D
and as a consequence, we have
Proposition 18. If D1 < 0, D2 < 0, D = D1 D2 are discriminants of quadratic
fields over Q and if√h1 , h2 are√the class numbers of and w1 , w2 the number of
roots of unity in Q( D1 ), Q( D2 ) respectively, then we have
X
4h1 h2
χ(B)G(B).
=Λ=
w1 w2
B
(130)
√
(In (130), χ is the genus character in Q( D) corresponding to the √
decomposition D = D1 D2 and B runs over all the narrow ideal classes in Q( D)).
Applications to Algebraic Number Theory
133
Remark. Formula (130) is interesting in that it gives another arithmetical significance for Λ. Besides, it is the analogue of Kronecker’s solution of Pell’s
equation which we referred to in § 3, Chapter II earlier. It has not appeared in
literature so far.
From (129) and (130), we have
H=
4hh1 h2 W r
.
w1 w2 w R
(131)
If we exclude the special cases when −4 ≤ D1 , D2 < 0, then w1 = w2 = 2 = w.
Moreover, r/R is 1 or 12 , as we have seen earlier. Further if we exclude F from
being
√ of the 8th or 10th or 12th roots of unity or the fields
√ the√cyclotomic√fields
Q( D, −1) or Q( D, −3), then W = 2. Thus barring these special cases,
we have for the class number H of F the formula
r
H=h Λ
R
150
and from (131)
r
hh1 h2 .
R
In general, R/r = 2 and hence we have, except for some special cases,
H=
2
(132)
H
= Λ = h1 h2 .
h
For D1 = −4, formula (131) was obtained by Dirichlet and in the general case, this was discovered by Hilbert. A generalization of (132) has been
considered by Herglotz. One breaks up√D as the product
of r(≥ 2) mutually co√
D
,
.
.
.
,
D
)
is
an unramified abelian
prime discriminants.
The
field
K
=
Q(
1
r
√
extension of Q( D) and for the Dedekind zeta function of F, we have the decomposition
as a product of L-series for the corresponding genus-characters in
√
Q( D). One then proceeds as above to obtain the √
required generalization
of
√
(132), involving the class numbers h1 , . . . hr of Q( D1 ), . . . Q( Dr ) respectively.
6 Some Examples
This section is devoted to giving a few interesting examples pertaining to the
determination of the class number of totally complex biquadratic extensions of
Q with particular reference to Proposition 17 and 18.
Examples 1-6 deal with the case when F is a totally complex number field
151
Applications to Algebraic Number Theory
134
√
which is an unramified imaginary
√
√ quadratic extension of Q( D), D > 0(f =
(1)). In fact, then F = Q( D1 , D2 ) where D = D1 D2 is a decomposition of
D as product of coprime negative discriminants D1 , D2 . Formula (129) gives
the class number H of F, on computing R, r, h, w, W and G(B). Incidentally
we also check (130), by finding h1 , h2 , w1 , w2 .
On the other hand we know that for any algebraic number field, the class
number can be found directly, as follows. Since,
√ in every class of ideals, there
exists an integral ideal of norm not exceeding |∆| (∆ being the absolute discriminant), it suffices
to test for mutual equivalence, the integral ideals of norm
√
not exceeding |∆| and obtain a maximal set of inequivalent ideals from among
these. This will give us the class number.
Our notation here will be the same as in the last section.
The following remark is to effect a simplification of the computation,
in the
√
case of Examples 1-6. The ray classes modulo f(= (1)) in Q( D) are just the
narrow ideal classes. Looking at the definition of G(B), we see that for two
narrow classes B1 and B2 lying in the same wide class, G(B1 ) = −G(B2 ) and
further χ(B1 ) = −χ(B2 ) so that χ(B1 )G(B1 ) = χ(B2 )G(B2 ). Thus
X
Λ=2
χ(B)G(B) = 2Λ∗ (say)
A
√
where A runs over the wide classes in Q( D) and B is a narrow class contained
in A. From (129), we have
H = 2h ·
r W
·
· Λ∗ .
R w
Further, formula (130) becomes
Λ∗ =
2h1 h2
.
w1 w2
√
As before, ǫ(> 1) is the generator of the group Γ∗f in Q( D).
Example 1. D = 12, D1 = −3, D2 = −4
√
√
√
h = 1, since (2) = (1 + 3)(1 − 3), (3) = ( 3)2 .
Similarly h1 = 1, h2 = 1. Further w1 = 6, w2 = 4, w = 2, ǫ = 2 +
√ 2

 1 + 3 
 , r/R = 1.
W = 12. Since ǫ = i 
1+i
√
3 > 0,
152
Applications to Algebraic Number Theory
135
√
Since h = 1, Λ∗ = G(B1 ),√ B1 being the principal
narrow class in Q( 12).
√
We take bB1 = (1); bB1 = [1, 3], β1 = 1, β2 = 3, v(β1 ) = 1.
√ !
! √ !
√
3
3
2 3
=
a = 2, b = 3, c = 1, d = 2.
(2 + 3)
1 2 1
1
1
1 2+2 1
·
− =
.
G(B1 ) =
12
1
4 12
Thus
H = 2.1.1 ·
Further
12 1
·
= 1.
2 12
2h1 h2 2.1.1
1
=
=
= Λ∗ .
w1 w2
6.4
12
Example 2. D = 24, D1 √
= −3, D2 = −8, h = h1 = h2 = 1, w = 2, w1 = 6,
w2 = 2, W = 6, ǫ = 5 + 2 6 > 0.
√ 2
√
∗
Since −ǫ = ( −3 + −2)
√ , r/R = 1. Also Λ √= G(B1 ), B1 being √the
principal narrow class in Q( 24); bB1 = (1) = [1, 6], β1 = 1, β2 = 6,
v(β1 ) = 1.
√ !
! √ !
6
6
5 12
ǫ
=
, a = 5, b = 12, c = 2, d = 5,
2 5
1
1
1 5+5 1 1
·
− = ,
Λ∗ = G(B1 ) =
12
2
4 6
6 1
H = 2.1 · · − 1,
2 6
2h1 h2 2.1.1
=
= Λ∗ .
w1 w2
6.2
Example 3. D = 140, D1 √= −4, D2 = −35, h1 = 1, w1 = 4, h2 = 2, w2 = 2,
h = 2, w = 2,
√ ǫ = 6 + 35 > 0, W = 4, r/R = 1/2. We take p√1 = (1),
p2 = (2, 1 + 35) as representatives of the two wide classes in Q( 35) and
denote the ray classes containing
them by B1 , B2 respectively. Since p22 = (2),
= −1.
N(p2 ) = 2 and χ(B2 ) = −35
2
√
For B1 , bB1 = p1 , β1 = 2, β2 = 35, v(β1 ) = 1, a = 6, b = 35, c = 1, d = 6,
G(B1 ) = 1/12 · (6 + 6)/1 − 1/4 = 3/4. √
√
For B2 , bB2 = p2 , β1 = 2, β2 = 1 + 35, v(β1 ) = 1, ω = (1 + 35)/2,
ǫ = 2ω + 5, ǫω = 7ω + 17, a = 7, b = 17, c = 2, d = 5,
153
Applications to Algebraic Number Theory
G(B2 ) =
136
1 7+5 1 1
·
− = ,
12
2
4 4
Λ∗ =
X
χ(B)G(B) =
A
3 1 1
− = ,
4 4 2
1 4 1
· · = 2,
2 2 2
2h1 h2 2.1.2
=
= Λ∗ .
w1 w2
4.2
H = 2.2 ·
Example 4. D = 140, D1 = −7, D2 = −20,√h1 = 1, w1 = 2, h2 = 2, w2 = 2,
h = 2, w = 2, W = 2, r/R = 12 , ǫ = 6 + 35 > 0. Let p1 , p2 , B1 , B2 have
!
−7
= 1, G(B1 ) = 3/4,
the same significance as in Example 3. Then χ(p2 ) =
2
∗
G(B2 ) = 1/4, Λ = 3/4 + 1/4 = 1,
1
· 1.1 = 2,
2
2h1 h2 2.1.2
=
= Λ∗ .
w1 w2
2.2
H = 2.2 ·
Example 5. D = 21, D1 √
= −3, D2 = −7, h1 = 1, w1 = 6, h2 =√1, w2 =
√ 2, h = 1,
w = 2, W = 6, ǫ = (5 + 21)/2 > 0; r/R = 1, since −ǫ = (( −3 + −7)/2)2 .
√
For the principal narrow class B1 in Q( 21), we take bB1 = (1) with integral basis [1, ǫ]; β1 = 1, β2 = ǫ, ω = ǫ, ǫω = 5ω − 1, a = 5, b = −1, c = 1,
d = 0.
1 5 1 1
· − = ,
12 1 4 6
6 1
H = 2.1.1 · · = 1,
2 6
2h1 h2 2.1.1
=
= Λ∗ .
w1 w2
6.2
Λ∗ = G(B1 ) =
154
We might show that H = 1, directly. The discriminant of F is (21)2 and we
have therefore to examine the splitting of (2), (3), (5), (7),
√ (11), (13), (17), (19)
in F. The ideals (2),
(5),
(11)
and
(17)
stay
prime
in
Q(
3) and (3), (13), (19)
√
√
√
stay prime in Q( −7). Further (7) = ( 7)2 in Q( −7). The further splitting
in F is seen to be as follows:
√ 
√ 

 1 + −7   1 − −7 
(2) = 
 
 ,
2
2
Applications to Algebraic Number Theory
137
√
(3) = ( −3)2 ,
√
√
√
√
(5) = (2 −3 + −7)(2 −3 − −7),
√
√
−3 + −7
2
−1 −2
(7) = (1 + ρ) (1 − ρ ) with ρ =
,
2
√
√
(11) = (2 + −7)(2 − −7),
√
√
(13) = (1 + 2 −3)(1 − 2 −3),
√  √
√ 
 √
 5 −3 + −7   5 −3 − −7 
(17) = 
 
 ,
2
2
√
√
(19) = (4 + −3)(4 − −3).
Thus all prime ideals of norm ≤ 21 are principal in F and hence H = 1.
Example 6. D = 2021, D1 = −43, D2 = −47, h1 = 1, w1 = 2.
!
√
−47
= −1 and so (5) stays prime in Q( −47),
To find h2 , we observe that
5
while (2) = p2 p′2 , (3) = p3 p′3 where
√


 1 + −47 
 ,

p2 = 2,
2
√ 

 1 ± 47 
 .

p3 = 3,
2
√
√
The integral ideals of norm ≤ |47| in Q( 47) are p2 , p′2 , p3 , p′3 , p22 , p′ 22 , p2 p3 ,
p′2 p3 , p2 p′3 , p′2 p′3 . Now
√


 1 + −47 
 = 12 = p22 p′ 22 p3 p′3 .
N 
2
Let p3 be so chosen that
√


 1 ± −47 
2 ′

 .
p2 p3 = 
2
Then p22 ∼ p3 (equivalent in the wide sense). Let now
√


 x + y −47 
 ,
pk2 = 
2
k being the order of the wide class containing p2 and x, y being rational integers.
Then x2 + 47y2 = 2k+2 and certainly y , 0, for then we would have pk2 = p′ k2
155
Applications to Algebraic Number Theory
138
i.e. p2 = p′2 , which is not true. But now the smallest k > 0 for which the
diophantine equation x2 + 47y2 = 2k+2 √
is solvable
with y , 0 is k = 5 (in



9
+
−47


2
4
fact, 92 + 47 = 128). Hence p52 = 
 and p2 / (1). Since p2 ∼ p3 ,
2
p′ 22 ∼ p′3 , p2 p3 ∼ p32 ∼ p22 ∼ p′3 , p′2 p3 ∼ p2 , p2 p′3 ∼ p′2 , p′2 p′3 ∼ p3 , we see that
(1), p2 , p′2 , p3 , p′3 form a complete set of mutually inequivalent integral ideals
√
of norm ≤ |47| and thus h2 = 5.
√
To find h, we have to consider integral ideals of norm ≤ 2021 (i.e. ≤ 44).
Since
!
2021
= −1 for p = 2, 3, 7, 11, 13, 23, 29, 31, 37, 41,
p
√
to consider therethese rational primes stay prime also in Q( 2021). We need √
fore only the decomposition of (5), (17), (19) and (43) in Q( 2021). Now
√
√



!
 43 + 2021   43 − 2021 
2021
= 0 and (43) = 
 
 .
43
2
2
Further
!
!
!
2021
2021
2021
=
=
=1
5
17
19
and so (5) = p5 · p′5 , (17) = p17 · p′17 , (19) = p19 · p′19 (say). Actually
Now,
√


 1 + 2021 
 .
p5 = 5,
2
√
√




 39 − 2021

 39 − 2021 
2
3
 .
p5 = 
, −25 and p5 = 
2
2
Hence p35 ∼ (1) but p25 / (1), for this would mean p5 ∼ (1) which is not true,
√
since ±5 is not the norm of any integer in Q( 2021). Now
√
√
(44 + 2021)(44 − 2021) = (5)(17)
and
(46 +
√
√
2021)(46 − 2021) = (5)(19)
and by properly choosing p17 , p19 , we can suppose that pp′17 − ∼ (1) and
p5 p′19 ∼ (1). But 1, p5 , p25 are inequivalent and p17 ∼ p5 ∼ p19 , p′17 ∼ p′19 ∼ p25 .
156
Applications to Algebraic Number Theory
139
√
Hence h = 3. Further w = 2, ǫ = 12 (45 + 2021); w2 = W = 2. Moreover,
r/R = 1 since
√
2
√
 −43 + −47 
−ǫ = 
 .
2
√
Let B1 , B2 , B3 be the narrow classes in Q( 2021) containing (1), p5 , p25 .
Then
√
√
1 + 2021
β2 1 + 2021
bB1 = (1), β1 = 1, β2 =
, ω1 =
=
,
2
β1
2
v(β1 ) = 1,

ǫω1 = 23ω1 + 505


 a = 23, b = 505, c = 1, d = 22
ǫ = ω1 + 22
1+
√
2021
, v(β1 ) = 1,
2
bB2 = p5 , β1 = 5, β2 =
√
β2 1 + 2021
ω2 =
=
,
β1
10

ǫω2 = 23ω2 + 101

 a = 23, b = 101, c = 5, d = 22
ǫ = 5ω2 + 22 
√
1
bB3 = p25 , β1 = (39 − 2021), β2 = −25, v(β1 ) = −1,
2
19 + ω1
−50
β2
=
=
,
ω3 =
√
β1 39 − 2021
5

ǫω3 = 42ω3 + 25

 a = 42, b = 25, c = 5, d = 3
ǫ = 5ω3 + 3 
!
−43
= −1, χ(B3 ) = χ(p25 ) = 1.
χ(B1 ) = 1, χ(B2 ) = χ(p5 ) =
5
Thus
(
)
1 23 + 22 1
Λ =1
−
12
1
4


!
!
4



23k
1
k

 1 23 + 22 X
P1
− 
+
−
P1
− 1



 12
5
5
5
4
k=1
∗
157
Applications to Algebraic Number Theory
Since P1
23k
5
140


!
!
4



k
42k
1

 1 42 + 3 X
P1
P1
− 
.
−
+



 12 5
5
5
4
k=1
= −P1
42k
5
, we see easily that Λ∗ = 5/2,
H = 2.3.1 ·
2 5
· = 15,
2 2
2h1 h2 2.1.5
=
= Λ∗ .
w1 w2
2.2
The following two examples deal√with the case when F is ramified (with
relative discriminant f , (1)) over Q( D), (cf. Proposition 17).√ Now the numbers γ and uB , vB corresponding to the ray classes B in Q( D) have to be
reckoned with.
√
√ √
Example 7. D = 5, F = Q( 5, θ) where θ = ( 5 − 5)/2
√ and −θ > 0,
5) are the √
same,
h = 1, w = 2, W
=
10.
The
fundamental
units
in
F
and
Q(
√
viz. ω √
= (1 + 5)/2 and so r/R = 1/2. It is easy to verify that θ = 5ω′ , 158
θ′ = − 5ω = θ · ω2 , N(ω) = −1.
√
√
If we set ρ = ( θ−ω)/2, then ρ−1 = −( θ+ω)/2 = −ρ−ω and so ρ+ρ−1 =
−ω. Therefore ρ2 + ρ−2 = ω2 − 2 = ω − 1, so that ρ2 + ρ−2 + ρ + ρ−1 + 1 = 0.
Actually, since Re ρ <√0 and Im ρ > 0, we have ρ = e4πi/5 . As a basis of the
integers √
in F over Q( 5), we √
have [1, ρ] and the relative discriminant f of F
over Q( 5) is (θ) = (θ′ ) = ( 5). Now ϕ(f) = 4 and√ω4 , −ω, ω, ω2 serve
as prime residue class representatives modulo f in Q( 5). In√fact, ω4 ≡ 1,
−ω ≡ 2, ω ≡ 3, ω2 ≡ 4( mod f). For ǫ, we take ω4 = (7 + 3 5)/2 and then
ǫ > 0 as also ǫ ≡ 1( mod f). We note that v(f)
√ = 0.
We take γ = 1/5 so√ that (γ)f = (1)/( 5) has exact denominator f and
q = (1). Further χ(q)v(γ 5) = 1 × −1 = −1.
In the notation of § 5 (Chapter II), the set {λk } consists just of 1, and {µl }
consists of 1, 2ω. Therefore, since h = 1, we may take as ray class representatives modulo f, the ideals p1 = (1) and p2 = (2) and denote the corresponding
ray classes by B1 and B2 .
For B1 , χ(B1 ) = 1, β1 = 1, β2 = ω, v(β1 ) = 1, a = 5, b = 3, c = 3, d = 2,
uB1 = S (γ) = 2/5, vB1 = S (ωγ) = 1/5.
For B2 , χ(B2 ) = −1, necessarily; β1 = 2, β2 = 2ω, v(β1 ) = 1, a = 5, b = 3,
c = 3, d = 2, uB2 = S (2γ) = 4/5, vB2 = S (2ωγ) = 2/5;

!
!
!
2



5(k + 2/5) 1 
1 5+2 X
k + 2/5

 (2/5)2 − (2/5)
Λ=
P1
−
+
−
− 
P1



2
12
3
3
3
5 
k=0
Applications to Algebraic Number Theory
141

!
!
!
2



k + 4/5
5(k + 4/5) 2 
1 5+2 X

 (4/5)2 − (4/5)
P1
P1
+
−
− 
−




2
12
3
3
3
5
k=0
= −2/5;
r W
1 10 2
H =h· ·
× (−Λ) = 1 · ·
· = 1.
R w
2 2 5
We shall now show that H = 1, directly. Since the absolute discriminant of
F is 52 · N(f) = 125, we need only to
! examine the splitting in F of the ideals
5
= −1 for p = 2, 3, 7 the ideals (2), (3),
(2), (3), (5), (7) and (11). Since
p!
√
θ
(7) are prime in Q( 5). Further,
= −1 for p = (2), (3), (7) and hence, by
p
(125), these stay prime in F too. Again,
√
√
√
√
√
√
(11) = (4 + 5)(4 − 5) = (1 − θ)(1 + θ)(1 − ω θ)(1 + ω θ)
√
√
(5) = ( 5)2 = ( θ)4 .
Thus, all prime ideals of norm ≤ 11 in F are principal and, as a consequence,
H = 1.
We might show that H = 1, also by using the following decomposition, due
to Kummer, of the Dedekind zeta-function ζ(s, F), viz.
ζ(s, F) = ζ(s)L5 (s)P(s)Q(s),
where, for σ > 1,
!
∞
X
5 −s
L5 (s) =
n ,
n
n=1
P(s) =
∞
X
ψ(n)n−s ,
Q(s) =
∞
X
ψ(n)n−s ,
n=1
n=1
and ψ(n) is defined by



0,






i,




ψ(n) = 
−1,





−i,





1,
n ≡ 0( mod 5),
n ≡ 2( mod 5),
n ≡ −1( mod 5),
n ≡ −2( mod 5),
n ≡ 1( mod 5).
159
Applications to Algebraic Number Theory
142
√
√
Since ζ(s, F) = ζ(s, Q( 5))L(s, χ) and ζ(s, Q( 5)) = ζ(s)L5 (s), we have
L(s, χ) = P(s)Q(s). Hence
P(1)Q(1) = L(1, χ) = −
π2 Λ 2π2
=
.
5
25
(∗)
But it is easy to see that
160
P(1) =
∞
X
n=−∞
!
i
1
,
+
5n + 1 5n + 2
π
(cot π/5 + i cot 2π/5),
5
!
π −i(ρ + ρ−1 ) ρ2 + ρ−2
+
.
=
5
ρ − ρ−1
ρ2 − ρ−2
=
Similarly
Q(1) =
!
π −i(ρ + ρ−1 ) ρ2 + ρ−2
.
−
5
ρ − ρ−1
ρ2 − ρ−2
Therefore
π2
P(1)Q(1) = −
25
π2
=−
25
π2
=−
25
π2
=−
25
!
ρ2 + ρ−2 + 2 ρ4 + ρ−4 + 2
+
ρ2 + ρ−2 − 2 ρ4 + ρ−4 − 2
!
ρ2 + ρ−2 + 2 ρ + ρ−1 + 2
+
ρ2 + ρ−2 − 2 ρ + ρ−1 − 2
!
α2 (α − 2) + (α + 2)(α2 − 4)
(α = ρ + ρ−1 = −ω)
(α2 − 4)(α − 2)
10 2π2
×−
=
,
5
25
which confirms (∗) above.
√
√
√
Example 8. D = 17, F =√Q( 17, θ) where θ = ( 17 − 5)/2, −θ > 0,√h = 1,
w = 2, W = 2; ρ = 4 + 17(= 2θ + 9) is the fundamental unit in Q( 17) as
also in F and so r/R = 1/2.
√
√
[1, (1 + θ)/θ′ ] is a basis
√ of the 3integers in F over Q( 17). The relative
ϕ(f) = 4
discriminant f of F over Q( 17) is (θ ); N(f) = 8 and v(f) = 0. Further
√
and as representatives of th prime residue classes modulo f in Q( 17), we take
1, −1, 3, −3. It is easy to verify that ρ ≡ −3, ρ2 ≡ 1, −ρ ≡ 3, −ρ2 ≡ −1(
mod f) and so, in every prime residue class modulo f, there is a unit. Also, it is
161
Applications to Algebraic Number Theory
143
√
√
clear that ǫ = ρ2 = 33 + 8 17 is the generator of Γ∗f . We take γ = 1/(θ3 17);
√
√
then (γ)( 17) = f−1 . So q = (1) and moreover, N(γ 17) > 0.
In the notation of § 5 (Chapter II) again, {γk } consists of the single element
1 while {µ1 } consists of the numbers 1, −3ρ, 5ρ, −7ρ2 . Further h = 1 and there
are just 4 ray classes modulo f which we denote by B1 , B3 , B5 and B7 (say) and
the corresponding representatives bB1 , bB3 , bB5 , bB7 may be chosen to be (1),
(3), (5), (7) respectively. The numbers uB , vB corresponding to the ray classes
B1 , B3 , B5 , B7 may be denoted by u1 , v1 , u3 , v3 , u5 , v5 , u7 , v7 respectively.
If λ ≡ ±1( mod f), then χ((λ)) = ±1 according as N(λ) ≷ 0 and hence
χ(B1 ) = 1, χ(B3 ) = χ((−3ρ)) = −1, χ(B5 ) = χ((5ρ)) = −1,!χ(B7 ) = χ((−7ρ2 )) =
2
.
1. We note incidentally that, for q = 1, 3, 5, 7, χ(Bq ) =
q
For B1 ,
bB1 = (1), β1 = 1, β2 = θ, u1 = S (γ) = −
v1 = S (θγ) ≡
23
,
8
1
( mod 1).
4
For B3 ,
bB3 = (3), β1 = 3, β2 = 3θ, u3 = S (3γ) = −
v3 = S (3θγ) ≡
69
,
8
3
( mod 1).
4
For B5 ,
bB5 = (5), β1 = 5, β2 = 5θ, u5 = S (5γ) = −
v5 = S (5θγ) ≡ −v3 (
115
,
8
mod 1).
For B7 ,
bB7 = (7), β1 = 7, β2 = 7θ, u7 = S (7γ) = −
v7 = S (7θγ) ≡ −v1 (
161
,
8
mod 1).
For all the 4 classes, a = −7, b = −32, c = 16, d = 73.
From (129), we now obtain
X
√
χ(Bq )G(Bq )
2H = χ((1))v(γ 17)
q=1,3,5,7
162
Applications to Algebraic Number Theory
144
i.e.
2H =
X
q=1,3,5,7
! 
!
!
15




k + uq
k + uq
33 X
2


7
·
×
P
.
P
(u
)
+
+
v
P

1
2
q
q
1




q
8
16
16
k=0
It may be verified that
X
f rac338
q=1,3,5,7
15
X
P1
15
X
P1
k=0
k=0
!
2
66
;
P2 (uq ) =
q
128
!
!
k + u1
k + u1
P1 7 ·
+ v1 =
16
16
!
!
15
X
k + u7
k + u7
6256
=
P1
P1 7 ·
+ v7 =
;
16
16
1282
k=0
!
!
k + u3
k + u3
+ v3 =
P1 7 ·
16
16
!
!
15
X
k + u5
k + u5
5904
P1
=
P1 7 ·
+ v5 = −
.
16
16
1282
k=0
Thus
2H =
6256 + 5904
66
+2
=2
128
1282
i.e. H = 1.
We shall now show that H = 1 directly. Since the absolute discriminant of
F is 172 × 8 = 2312(< 492 ), we have to examine for mutual equivalence all
integral ideals of (absolute) norm < 49 in F.
To this end, we first remark√that the ideals (3), (5), (7), (11),
√ (23), (29), (31),
(37) and (41) are prime in Q( 17). Further we have in Q( 17) the following
decomposition of other ideals, viz.
(2) = (θ)(θ′ ), (13) = (1 + 4θ′ )(1 + 4θ), (17) = (5 + 2θ)(5 + 2θ′ ),
(19) = (11 + 2θ)(11 + 2θ′ ), (43) = (1 + 6θ)(1 + 6θ′ ),
(47) = (3 − 2θ)(3 − 2θ′ ).
√
Now, if p is a prime ideal in Q( 17), then


! 
+1, if p = B1 B2 in F, B1 , B2



θ

=
χ(p) =
−1, if p is prime in F


p


0,
if p = B2 in F.
163
Applications to Algebraic Number Theory
145
√
Further, if λ !and µ are !integers in Q( 17) such that λ ≡ µ( mod f), then
θ
θ
=±
according as N(λµ) ≷ 0.
χ((λ)) =
(λ)
(µ)
!
√
θ
For p = (3), (5), (11), (29), (37) in Q( 17),
= −1 and so these prime
p
ideals stay prime also in F.
√ 
√ 

√ 2  1 + θ   1 − θ 
′



 .
(2) = (θ)(θ ) = ( θ) 
 
θ′
θ′
Now 1+4θ′ ≡ 3( mod f) and 1+4θ = 1( mod f) so that χ((1+4θ′ )) = −1,
while χ((1 + 4θ)) = +1. So (1 + 4θ′ ) stays prime in F while (1 + 4θ) splits in
F. This gives
√
√
(13) = (1 + 4θ′ )(1 + θ + θ)(1 + θ − θ).
Again 5 + 2θ ≡ 1( mod f), 5 + 2θ′ ≡ −1( mod f) and so χ((5 + 2θ)) =
χ((5 + 2θ′ )) = −1. Hence the ideal (17) does not decompose further in F.
In view of the fact that 11 + 2θ′ ≡ −3( mod f), χ((11 + 2θ′ )) = −1 and
so (11 + 2θ′ ) stays prime in F whereas (11 + 2θ) splits in F. We have in F
therefore the decomposition
√
√
(19) = (11 + 2θ′ )(1 + (5 + θ) θ)(1 − (5 + θ) θ).
164
From 1 + 6θ′ ≡ −1( mod f), 1 + 6θ ≡ −3( mod f) we know that (1 + 6θ)
stays prime in F while (1 + 6θ′ ) splits and we obtain the splitting of (43) in F
as
√
√
(43) = (1 + 6θ)(5 + θ + θ(4 + θ))(5 + θ − θ(4 + θ)).
Both the ideals (3 − 2θ) and (3 − 2θ′ ) split in F and we have
√
√
√
(47) = (1 + θ(3 + θ))(1 − θ(3 + θ))(3 + θ + θ(4 + θ))×
√
× (3 + θ − θ(4 + θ)).
√
The prime ideals (7) and (41) in Q( 17) split in F as
√
√
(7) = (θ + 3 + θ)(θ + 3 − θ),
√
√
(41) = (3 + 2θ + 2 θ(5 + θ))(3 + 2θ − 2 θ(5 + θ)).
Finally, from the decompositions
√
√



√   (8 + 3θ) + θ(θ − θ)/2 
 8 + 3θ
θ
′
 ,


(θ + θ) 
(23)(θ ) =  √ +
√
2
θ
θ
Applications to Algebraic Number Theory
√ 
√ 


θ2 θ θ 
θ2 θ θ  
 −10 − 7θ +
 ,
+
−
(62)(1 + 4θ) = −10 − 7θ +
2
2
2
2
146
we can deduce that the ideals (23) and (31) split into two distinct principal
prime ideals each in F, if we use the decomposition of (1 + 4θ) and (2) in F.
Thus all prime ideals of norm < 49 in F are principal and we have H = 1.
Bibliography
[1] R. Dedekind: Aus Briefen an Frobenius, Gesammelte mathematische
Werke, Zweiter Band, Braunschweig, (1931), 420.
[2] M. Deuring: Die Klassenkörper der komplexen Multiplikation, Enzykl
der math. Wiss., Band I, 2, 23, Teubner, (1958).
[3] P.G.L. Dirichlet: Recherches sur les formes quadratiques à coefficients
et à indéterminées complexes, Werke, Bd. I, 533-618.
[4] R. Fricke: Lehrbuch der Algebra, Vol. 3, Braunschweig, (1928).
[5] R. Fueter: Die verallgemeinerte Kroneckersche Grenzformel und ihre
Anwendung auf die Berechnung der Klassenzahl, Rendiconti del circ.
Mathematico di Palermo, 29 (1910), 380-395.
[6] C.F. Gauss: Disquisitiones Arithmeticae, Gesammelte Werke 1, (1801).
[7] H. Hasse: Bericht über neuere Untersuchungen und Probleme aus der
Theorie der algebraischen Zahlkörper I, Jahresbericht der D. M. V. 35
(1926) 1-55.
[8] E. Hecke: Über die Zetafunktion beliebiger algebraischer Zahlkörper,
Werke, 159-171.
[9] E. Hecke: Über eine neue Anwendung der Zetafunktionen auf die Arithmetik der Zahlkörper, Werke, 172-177.
[10] E. Hecke: Über die Kroneckersche Grenzformel für reelle quadratische
Körper und ide Klassenzahl relativ-Abelscher Körper. Werke, 198-207.
[11] E. Hecke: Eine neue Art von Zetafunktionen und ihre Beziehungen zur
Verteilung der Primzahlen. I, II, Werke, 215-234, 249-289.
147
165
Bibliography
148
[12] E. Hecke: Bestimmung der Klassenzahl einer neuen Reihe von algebraischen Zahlkörpern. Werke. 290-312.
[13] E. Hecke: Darstellung von Klassenzahlen als Perioden von Integralen 3.
Gattung aus dem Gebiet der elliptischen Modulfunktionen. Werke, 405417.
[14] E. Hecke: Theorie der Eisensteinschen Reihen höherer Stufe und ihre
Anwendung auf Funktionentheorie und Arithmetik. Werke, 461-486.
[15] G. Herglotz: Über einen Dirichletschen Satz. Math. Zeit, 12 (1922), 255261.
[16] G. Herglotz: Über die Kroneckersche Grenzformel für reelle quadratische Körper. 1, 11, Ber. d. Sachs Akod. d. Wiss. zu Leipzig. 75 (1923).
[17] D. Hilbert: Bericht über die Theorie der algebraischen Zahlkörper,
Jahresbericht der D.M.V. (1897), 177-546. Ges. Werke. Bd. 1 (1932), 63363.
[18] D. Hilbert: Über die Theorie des relativquadratischen Zahlkörpers,
Math. Ann. 51 (1899), 1-127. Ges. Werke Bd 1 (1932), 370-482.
[19] L. Kronecker: Zur Theorie der Elliptischen Funktionen, Werke, Vol. 4,
347-495, Vol. 5, 1-132, Leipzig, (1929).
[20] E. Kummer: Mémoire sur la theorie des nombres complexes composés de
racines de l’unité et de nombres entiers, Jour. de math. pures et appliquees
XVI (1851), 377-498 (collected papers I, 363-484).
[21] H. Maass: Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141-183.
[22] C. Meyer: Die Berechnung der Klassenzahl abelscher Körper über
quadratischen Zahlkörpern, Akademie Verlag, Berlin, 1957.
[23] A. Selberg: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian
Math. Soc. 20 (1956), 47-87.
166
Chapter 3
Modular Functions and
Algebraic Number Theory
1 Abelian functions and complex multiplications
167
In this section, we shall see how the complex multiplications of abelian functions lead in a natural fashion to the study of the Hilbert modular group.
Let z1 , . . . , zn be n independent complex variables and let
 
z1 
 
z =  ... 
 
zn
Let C n be the n-dimensional complex euclidean space and G, a domain in C n .
A complex-valued function f (z) defined on G (except, perhaps, for a lowerdimensional subset of G) is meromorphic in G, if there exists a correspondence
 
a1 
 
pa (z)
a =  ...  (∈ G) →
qa (z)
 
an
where pa (z) and qa (z) are convergent power-series in z1 − a1 , . . . zn − an with a
pa (z)
common domain of convergence Ka around a such that for z ∈ Ka , f (z) =
qa (z)
whenever either qa (z) , 0 or if qa (z) = 0, the pa (z) , 0.
149
Modular Functions and Algebraic Number Theory
150
 p1 
 
Let f (z) be a meromorphic function of z in C . A complex column p =  ... 
 0  pn
 
is a period of f (z), if f (z + p) = f (z) for all z ∈ C n ; for example, 0 =  ...  is a
n
0
period of f (z). The periods of f (z) form an additive abelian group; if we regard
C n as a 2n-dimensional real vector space, then the group of periods is, in fact,
a closed vector group over the field of real numbers. Either this vector group
is discrete or it contains at least one limit point in C n . In the latter case, f (z)
is said to be degenerate; this is equivalent to the fact that f (z) has infinitesimal
periods and by a suitable linear transformation of the variables z1 , . . . , zn with
complex coefficients, the function f (z) can be brought to depend on strictly less
than n complex variables. In the former case, f (z) is said to be non-degenerate
and its period-group is a discrete vector-group i.e. a lattice in C n . We shall
consider only the case when the period-lattice has 2n generators linearly independent over the field of real numbers. It is easy to construct an example of
a non-degenerate abelian function with 2n independent periods. For example,
let P(w) be the Weierstrass’ elliptic function with independent periods ω1 ,
ω2 . Then the function f (z) = P(z1 ) . . . P(zn ) furnishes the required example,
since it has exactly 2n independent periods
168
   
       
 0   0 
ω1  ω2   0   0 
 ·   · 
 ·   ·  ω  ω 
   
     1   2 
 ·  ,  ·  ,  0  ,  0  , . . . ,  ·  ,  ·  .
 0   0 
 0   0   ·   · 
   
       
0
0
0
ω1 ω2
0
Let, then, for a given f (z) meromorphic in C n , p1 , . . . , p2n be 2n periods
linearly independent over the field of real numbers such that any period p of
P
f (z) is of the form 2n
i=1 mi pi with rational integers mi . The n-rowed matrix
P = (p1 , . . . , p2n ) is called a period-matrix of f (z). It is uniquely determined
upto multiplication on the right by a 2n-rowed unimodular matrix. We shall
denote the period-lattice generated by p1 , . . . , p2n in C n by L.
A meromorphic function f (z) defined in C n and admitting as periods all the
elements of L is called an abelian function for the lattice L. The abelian functions for L constitute a field FL . In FL there exists at least one function f (z)
having L exactly as its period-lattice. Moreover, in FL , there exist n functions
f1 , . . . , fn which are analytically independent and hence algebraically independent over the field of complex numbers. Also, every abelian function satisfies a
polynomial equation of bounded degree with rational functions of f1 , . . . , fn as
coefficients. As a consequence, it can be shown that FL is an algebraic function
169
Modular Functions and Algebraic Number Theory
151
field of n variables and one can find n + 1 functions, f0 , f1 , . . . , fn in FL such
that every function in FL is a rational function of f0 , f1 , . . . , fn with complex
coefficients.
Let P be a complex matrix of n rows and 2n columns such that the 2n
columns are linearly independent over the field of real numbers. The problem
arises as to when P is a period-matrix of a non-degenerate abelian function or,
in other words, a Riemann matrix. It is well known that in order that P might
be a Riemann matrix, it is necessary and sufficient that there exists a rational
2n-rowed alternate (skew-symmetric) matrix A such that
(i) PAP′ = 0,
(ii) H = i−1 PAP′ > 0 (i.e. positive hermitian).
(133)
These are known as the period relations.
IfE is the n-rowed identity matrix
0 E , then conditions (i) and (ii) were
and A is the 2n-rowed alternate matrix −E
0
given by Riemann as precisely the conditions to be satisfied by the periods of a
normalised complete system of abelian integrals of the first kind on a Riemann
surface of genusn. P
, the two conditions above may be written in the form
Setting B =
P
!
0 −iH
′
BAB =
, H > 0.
(134)
iH
0
For n = 1, the conditions (133) reduce to the sole condition that if P = (ω1 ω2 ),
then ω1 ω−1
2 should not be real. To establish the necessity and sufficiency of
these conditions for n > 1, one has to make considerable use of theta-functions.
The problem as to when a given P is a Riemann matrix was apparently
considered independently also by Weierstrass and he wished to prove to this
end that every abelian function can be written as the quotient of two thetafunctions. He could not, however, complete the proof of this last statement–it
was Appell who gave a complete proof of the same for n = 2, and Poincaré,
for general n.
A period matrix P which satisfies the period-relations with respect to an
alternate matrix A is said to be polarized with respect to A. The alternate matrix A is not unique; but, in general, A is uniquely determined upto a positive
rational scalar factor. Moreover, if P is polarized with respect to A, then from
(134), since H > 0, we conclude that |A| , 0, |B| , 0.
Let P be a Riemann matrix and L, the associated lattice in C n . Let L1 be
a sublattice of L, again of rank 2n over the field of real numbers and let P1
be a period matrix of L1 . Then P1 = PR1 for a nonsingular rational integral
170
Modular Functions and Algebraic Number Theory
152
matrix R1 . The associated function field FL1 is an algebraic function field of
n variables, containing FL . Since FL itself is an algebraic function field of n
variables, we see that FL1 is an algebraic extension of FL .
Conversely, let FL1 be a field of abelian functions, P1 an associated Riemann matrix and L1 the corresponding period-lattice in C n . Further, let F be a
subfield of FL1 such that FL1 is algebraic over FL . Then F is again a field of
abelian functions containing at least one non-degenerate abelian function and
has a period-lattice L containing L1 . This means that if P is a period-matrix
associated with F, then P1 = PR1 for a nonsingular rational integral matrix R1 .
We now prove
Proposition 19. Let P1 and P2 be two Riemann matrices and L1 and L2 the
associated period lattices in C n . A necessary and sufficient condition that the
elements of FL1 depend algebraically on those of FL2 and vice versa, is that
P1 = PM for a nonsingular rational matrix M.
Proof. Let f0 , . . . , fn be generators of FL1 over the field of complex numbers
and let elements of FL1 depend algebraically on those of FL2 . This means that
each fi satisfies an irreducible polynomial equation
ni −1
fini + a(i)
+ · · · + a(i)
ni −1 fi
0 = 0,
where a(i)
m , m = 0, 1, . . . , ni − 1 belong to FL2 . Moreover, it is easy to see
that a(i)
m for m = 0, 1, . . . , ni − 1 and i = 0, 1, . . . , n lie in FL1 . Then the field
FL generated by {a(i)
m } over the field of complex numbers is a field of abelian
functions contained in both FL1 and FL2 and hence admitting for periods, all
the elements of L1 and L2 . Since FL1 is algebraic over FL , FL is an algebraic
function field of n variables and hence FL2 is also algebraic over FL . Let P be
a period matrix of FL and L, the associated lattice in C n . Then L contains both
L1 and L2 . Hence P1 = PR1 and P2 = PR2 for nonsingular rational integral
matrices R1 and R2 . Thus P1 = P2 M for a nonsingular rational matrix M. Conversely, if P1 = P2 M for a nonsingular rational matrix M, then there
exists a Riemann matrix P such that P1 = PR1 and P2 = PR2 for nonsingular
rational integral matrices R1 and R2 respectively. If L is the period-lattice in C n
associated with P, the field FL of abelian functions for L, is contained in both
FL1 and FL2 . Moreover FL1 and FL2 are algebraic over FL . Thus the condition
is also sufficient.
Two lattices L1 and L2 in C n , of rank 2n over the field of real numbers
and having P1 and P2 for period-matrices respectively are said to be commensurable if there is another lattice L of rank 2n containing both L1 and L2 or
171
Modular Functions and Algebraic Number Theory
153
equivalently: P1 = P2 M for a nonsingular rational matrix M. Thus in Proposition 19, the necessary and sufficient condition may also be stated that the
lattices L1 and L2 associated with P1 and P2 are commensurable.
Let f (z) be a non-degenerate abelian function with period matrix P and
period lattice L and let m be a scalar. The function f (mz) has period matrix
m−1 P. If m is rational, then the period lattice associated with m−1 P is clearly
commensurable with L and hence f (mz) depends algebraically on the field FL ,
in view of Proposition 19. This is the analogue of the well-known multiplication theorem for elliptic functions. We now ask for all scalars m such that if
f (z) ∈ FL , f (mz) is algebraic over FL . Let us first consider the case n = 1.
Let p1 , p2 be two independent periods of an elliptic function f (z) such that
every other period p of f (z) is of the form rp1 + sp2 with rational integral r
and s. In order that f (mz) be algebraic over the field of elliptic functions with
periods p1 and p2 , it is necessary and sufficient that
!
a b
m(p1 p2 ) = (p1 p2 )
c d
with rational a, b, c, d. This means that m is the root of a quadratic equation
with rational coefficients. More precisely, m should be an imaginary quadratic
irrationality lying in the field Q(p2 p−1
1 ) where Q is the field of rational numbers. If we require that f (mz) should again be an elliptic funciton with p1 and
p2 as periods, then a, b, c, d should be rational integers. This was the principle of complex multiplication for elliptic functions formulated by Abel in his
work, “Recharches sur les fonctions elliptiques”. The idea of complex multiplication in its simplest form, however, appears to be contained implicitly in
the work of Fagnano, concerning the doubling of an arc of the lemniscate of
Bernoulli.
If f (z) is a non-degenerate abelian function with period matrix P and period
lattice L and Q is an n-rowed complex non-singular matrix, then f (Q−1 z) is a
non-degenerate abelian function with period matrix QP. We formerly wished
to find all complex numbers m such that for every f (z) ∈ FL , f (mz) is algebraic
over FL . We now ask, more generally, for all n-rowed complex non-singular
matrices Q such that for every f (z) ∈ FL , f (Q−1 z) is algebraic over FL . By
Proposition 19, a necessary and sufficient condition to be satisfied by Q is that
QP = PM
(135)
for some rational non-singular 2n-rowed matrix M. The matrix M is called
a multiplier of P and Q, a complex multiplication of P. Trivial examples of
multipliers are λE, where λ ∈ Q and E = E (2n) is the 2n-rowed identity matrix.
172
Modular Functions and Algebraic Number Theory
154
From QP = PM, it follows that QP = PM, since M is rational. Hence
(135) is equivalent to the condition
"
#
Q 0
B = BM.
0′ Q
Since B is non-singular, we observe that for given M, Q is uniquely determined
by
#
"
Q 0
(136)
BMB−1 = ′
0 Q
and vice versa. The problem of determining all the complex multiplications Q
of a given Riemann matrix P reduces to that of finding all 2n-rowed rational
non-singular matrices M such that BMB−1 breaks up as in (136) for some nrowed complex non-singular Q.
We now drop the condition that M be non-singular and call M, a multiplier
still, if the matrix BMB−1 breaks up as in (136) or equivalently PM = QP for
some Q. We may denote by R, the set of all multipliers M of P. We see first
that R is a ring containing identity. For, if M1 , M2 ∈ R, then for some Q1 and
Q2 , we have Q1 P = PM1 and Q2 P = PM2 . Hence
P(M1 + M2 ) = (Q1 + Q2 )P,
PM1 M2 = Q1 PM2 = Q1 Q2 P
and thus M1 + M2 , M1 M2 ∈ R. Further the 2n-rowed identity matrix E lies in
R and serves as the identity in R. Since for M ∈ R and λ ∈ Q, λM also lies in
R, we see that R is an algebra over Q and in fact, of finite rank over Q.
′ −1
We now remark that if M ∈ R, M∗ = AM
A also lies in R. This is simple
Q 0
′
to verify. Let us set G = BAB , T = 0′ Q ; then

′ −1

GT ′G−1 = HQ ′H
0

0  .

HQ′ H −1
Further, since B′ T ′ = M ′ B′ and B′ = A−1 B−1G, we have A−1 B−1GT ′ =
M ′ A−1 B−1G i.e. (GT ′G−1 )B = B(AM ′ A−1 ). This proves that M ∗ ∈ R. The
mapping M → M ∗ has the following properties, viz. if M1 , M2 ∈ R, then
(M1 + M2 )∗ = A(M1′ + M2′ )A−1 = M1∗ + M2∗ ,
(M1 M2 )∗ = AM2′ M1′ A−1 = AM2′ A−1 AM1′ A−1 = M2∗ M1∗ ,
(M ∗ )∗ = (AM ′ A−1 )∗ = M.
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Modular Functions and Algebraic Number Theory
155
Thus the mapping M → M ∗ is an involution of R and is usually called the
Rosati involution. Further, if the matrix Q corresponds to the multiplier M ∈
′ −1
R, then to M ∗ corresponds the complex multiplication Q∗ = HQ H .
Thus R is an involutorial algebra of finite rank over Q. Such algebras have
been extensively studied by A. A. Albert who has also determined the involutorial algebras which can be realized as the rings of multipliers of Riemann
matrices. We shall not go into the general theory of such algebras. We are
interested only in involutorial algebras which can be realized as the ring of
multipliers of a Riemann matrix P and which, in addition, are commutative
rings free from divisors of zero. Such a ring of multipliers R is necessarily a
field and in fact an algebraic number field k of degree m say, over Q. So R
gives a faithful representation of k into M2n (Q), the full matrix algebra of order
2n over Q. Before we proceed further, we need to study this representation of
k by R more closely.
First, we have the so-called regular representation of k with respect to a
basis ω1 , . . . , ωm of k over Q. Namely, let for γ ∈ k, γ(i) , i = 1, 2, . . . , m denote
the conjugates of γ over Q. Then we have for l = 1, 2, . . . , m
γωl =
m
X
ωq cql ,
174
(137)
q=1
where cql ∈ Q. If we denote by C the matrix (cql ), 1 ≤ q, l ≤ m with q
and l respectively as the row index and column index, then the mapping γ →
C ∈ Mm (Q) is an isomorphism and gives the regular representation of k with
respect to the basis ω1 , . . . , ωm of k over Q. If we change the basis, we get
an equivalent representation. Hereafter, we shall always refer to the regular
representation of k with respect to a fixed basis ω1 , . . . , ωm of k over Q.
Now, from (??), we have
γ
( j)
ω(l j)
=
m
X
ω(qj) cql , j = 1, 2, . . . , m.
(137)∗
q=1
Let [γ] denote the m-rowed diagonal matrix [γ(1) , . . . , γ(m) ], with γ(1) , . . . , γ(m)
as diagonal elements. Further, let us denote by Ω, the m-rowed square matrix
(ω(i j) ) where i and j are respectively the column index and row index. Now
(137)∗ reads as
[γ]Ω = ΩC.
Or, since Ω is non-singular,
175
[γ] = ΩCΩ−1 .
(138)
Modular Functions and Algebraic Number Theory
156
Conversely, if G = [γ1 , . . . , γm ] is a diagonal matrix with diagonal elements
γ1 , . . . , γm such that Ω−1GΩ is a rational matrix C, then necessarily γi = γ(i) ,
i = 1, 2, . . . , m, for some γ ∈ k. In fact,
G = ΩCΩ−1 = ΩC(Ω′ Ω)−1 Ω′ .
Now, the matrix Ω′ Ω = (S (ωk ωl )) is rational and hence, also, the matrix
C(Ω′ Ω)−1 = (rkl ) is rational. thus
γk =
m
X
(k)
ω(k)
p r pq ωq
p,q=1
with r pq ∈ Q, for k = 1, 2, . . . , m. If we set
γ=
m
X
ω p r pq ωq
p,q=1
then we see that γi = γ(i) for i = 1, 2, . . . , m. We shall make use of this remark
later.
In the sequel, if C1 , . . . , Cr are square matrices then [C1 , . . . , Cr ] shall stand
for the direct sum of C1 , . . . , Cr .
We shall now prove a theorem concerning an arbitrary faithful representation of k into M2n (Q).
Theorem 14. Let k be an arbitrary algebraic number field of degree m over Q
and let ψ : γ → G ∈ M2n (Q) be faithful representation of k, of order 2n. Then
(i) 2n = qm, for a rational integer q ≥ 1 and
(ii) there exists a 2n-rowed non-singular rational matrix T independent of γ
such that
T −1GT = [C, . . . , C],
C being the image of γ under the regular representation of k.
Proof. Let γ ∈ k generate k over Q and let f (x) =
m
P
ai xi (am = 1) be the
i=0
irreducible polynomial of γ over Q. If ψ(γ) = G, we can find a complex nonsingular matrix W such that W −1GW is in the normal form, namely


γi 1 0 · · 0 
 0 γi 1 · · 0 


−1
· · · · 0 
W GW = [G1 , . . . , Gh ], Gi =  ·

 ·
· · · · 1 

0 · · · 0 γi
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Modular Functions and Algebraic Number Theory
157
P
i
and γ1 , . . . , γh are eigenvalues of G. Using the fact that a0 E (2n) + m
i=1 ai G = 0,
(2n) Pm
−1 i
we see that γi are conjugates of γ over Q. Again, from a0 E + i=1 ai W G W =
0, we conclude that W −1GW is necessarily in the diagonal form. For, otherwise, even if one Gi is of the form with the elements just above the main
m
P
ja j γij−1 = 0 contradicting the fact that
diagonal equal to 1, then we obtain
j=1
γi has an irreducible polynomial of degree m. Thus W −1GW = [γ1 , . . . , γ2n ]
where γ1 , . . . , γ2n are conjugates of γ over Q. Moreover, since the characteristic polynomial |xE − G| of G has rational coefficients and has for its zeros only
those of f (x), it follows that |xE − G| = ( f (x))q for some rational integer q.
Thus 2n = mq, which proves (i). Moreover γ1 , γ2 , . . . , γ2n are just the numbers
γ(1) , . . . , γ(m) repeated q times. We may therefore suppose that
W −1GW = [[γ], . . . , [γ]],
where [γ] = [γ(1) , . . . , γ(m) ]. But from (138), we have
W −1GW = V[C, . . . , C]V −1 ,
(139)
where V = [Ω, . . . , Ω].
Now, since 1, γ, . . . , γm−1 form a basis of k over Q, we have, in order to
prove (ii), only to find a rational matrix T such that
T −1GT = [C, . . . , C].
From (139), we see that there exists a non-singular complex matrix U = (ui j ),
1 ≤ i, j ≤ 2n such that
GU = U[C, . . . , C].
(140)
The matrix equation (140) may be regarded as a system of linear equations in
ui j with rational coefficients. Since there exists a solution in complex numbers
for this system of linear equations, we know from the theory of linear equations
that there exists at least one non-trivial solution for this system, in rational
numbers. The rational solutions of this system form a non-trivial vector space
of finite dimension over Q. Let U1 , . . . , Ur be the matrices corresponding to
a set of generators of this vector space. From the theory of linear equations
again, we see that all complex solutions U of (140) are necessarily of the form
z1 U1 + · · · + zr Ur with arbitrary complex z1 , . . . , zr . Now, we know that there
exists at least one complex solution U of (140) for which |U| , 0. This means
that the polynomial |z1 U1 + · · · + zr Ur | in z1 , . . . , zr does not vanish identically.
Hence we can find λ1 , . . . , λr ∈ Q such that |λ1 U1 + · · · + λr Ur | , 0. If we set
T = λ1 U1 + · · · + λr Ur , then |T | , 0 and
GT = T [C, . . . , C],
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Modular Functions and Algebraic Number Theory
158
i.e.
T −1GT = [C, . . . , C] = V −1 [[γ], . . . , [γ]]V.
Our theorem is therefore proved.
The theorem above gives us the exact form of the multipliers when the ring
of multipliers is an algebraic number field. With this preliminary investigation
regarding the ring of multipliers, we may now proceed to define the generalized
half planes and the general modular group.
Let A be a 2n-rowed non-singular rational alternate matrix. We denote by
BA , the set of all period matrices P which are polarized with respect to A. If
P ∈ BA , it is clear that, for n-rowed complex non-singular Q, QP ∈ BA .
Now, if A1 and A2 are two such alternate matrices, then we can find a rational non-singular matrix S such that A1 = S A2 S ′ . It is easy to verify that BA2
consists precisely of the period matrices of the form PS , for P ∈ BA1 . Thus
there is no loss of generality in confining ourselves to a fixed 2n-rowed
rational
0 E where
alternate matrix A with |A| , 0; we might take for A the matrix −E
0
E is the n-rowed identity matrix.
Denoting by R, the group of n-rowed complex non-singular matrices, we
know that if P ∈ BA , then for every Q ∈ R, QP ∈ BA . Now, we introduce an
equivalence relation in BA ; namely, two matrices P1 , P2 ∈ BA are equivalent
if P1 = QP2 , for Q ∈ R. The resulting set R\BA of equivalence classes is
denoted by HA and is called a generalized half-plane. It may be verified that
HA can be identified with the space of n-rowed complex symmetric matrices
Z = X + iY with Y > 0.
Let ΓA be the group of 2n-rowed unimodular matrices U such that UAU ′ =
A. If P ∈ BA , then for U ∈ ΓA , PU ∈ BA . On BA , the group ΓA acts in the
homogeneous manner; namely, if P = (FG) ∈ BA and U = AB CD ∈ ΓA where
A, B, C, D, F, G are n-rowed square matrices, then PU = (FA +GB FC +GD).
On HA , ΓA acts in the inhomogeneous way; namely, if Z ∈ HA and U = AB CD ∈
ΓA then U takes Z to (A + ZB)−1 (C + ZD). The group ΓA is the well-known
modular group of degree n. It has been shown by Siegel that ΓA acts as a
discontinuous group of analytic homeomorphisms of HA onto itself and that a
nice fundamental domain FA can be constructed for ΓA in HA . It is known that
FA is a so-called ‘complex space’.
Let R be the ring of multipliers of a Riemann matrix P polarized with
respect to A. Corresponding to M ∈ R, there is a Q uniquely determined by
(136). We denote this set of Q by D and shall, hereafter, refer to a Q ∈ D
corresponding to an M ∈ R.
For the given A and R, we denote by BA,R the set of all n-rowed Riemann
matrices P polarized with respect to A and having the property that for every
178
Modular Functions and Algebraic Number Theory
159
M ∈ R, PM = QP where Q ∈ D corresponds to M ∈ R. Clearly BA,R is not
empty, since it contains the Riemann matrix P we started with. Moreover, let
RR denote the group of K ∈ R such that QK = KQ for all Q ∈ D. It is obvious
that if P ∈ BA,R and K ∈ RR , then KP ∈ BA,R . We now introduce in BA,R
the equivalence relation that if P1 , P2 ∈ BA,R , then P1 and P2 are equivalent
provided that P1 = KP2 for K ∈ RR . The resulting set of equivalence classes
is denoted by HA,R and called the generalized half-plane corresponding to R.
It is to be noted that HA,R may also be defined as follows: If B∗A,R denotes the set of all n-rowed Riemann matrices P polarized with respect to A
and admitting for multipliers all elements of R, then we introduce in B∗A,R the
equivalence relation that P1 , P2 ∈ B∗A,R are equivalent if P1 = KP2 for K ∈ R.
The resulting set of equivalence classes can be easily identified with HA,R .
Let now ΓA,R be the subgroup of ΓA , consisting of unimodular U such that
U M = MU for all M ∈ R. We call ΓA,R the modular group corresponding to
R. Unless R consists of just the trivial multipliers λE where λ ∈ Q and E is
the 2n-rowed identity matrix, ΓA,R is, in general, much smaller than ΓA . Now,
if P ∈ BA,R and U ∈ ΓA,R , it is easy to see that PU ∈ BA,R . The group ΓA,R
acts on HA,R in the following way: namely, if U ∈ ΓA,R , then U takes the class
containing P ∈ BA,R into the class containing PU. It is known that in this way
ΓA,R acts discontinuously on HA,R and one can construct a fundamental domain
FA,R for ΓA,R in HA,R .
The space HA,R is the most general half-plane and the group ΓA,R is the
most general modular group of degree n. Modular groups of such general
type were first considered by Siegel. The modular group of degree n over a
totally real algebraic number field arose in a natural way in connection with the
researches of Siegel on the analytic theory of quadratic forms over algebraic
number fields. Later, the modular group of degree n over a simple involutorial
algebra was discussed by Siegel fully, in his work, “Die Modulgruppe in einer
einfachen involutorischen Algebra”.
As indicated earlier, we are interested in finding HA,R and ΓA,R only when
R is a field i.e. an algebraic number field k of degree m, say, over Q. We shall
see presently that we are led in a natural fashion to the Hilbert modular group,
when k is totally real.
First, we might make some preliminary simplification. We might suppose
that if M ∈ R corresponds to γ ∈ k, then M is of the form [C, . . . , C] where
C is the image of γ under the regular representation. In fact, by Theorem 14,
we can find a rational matrix T such that T −1 MT = [C, . . . , C] uniformly for
all M ∈ R. Let us now set A1 = T −1 AT ′ −1 and let R1 be the set of matrices of
the form T −1 MT for M ∈ R. Then R1 gives an equivalent representation of k
and all the matrices in R1 are of the desired form. Moreover, BA1 ,R1 consists
179
180
Modular Functions and Algebraic Number Theory
160
precisely of the matrices of the form PT where P ∈ BA,R . Hence we might as
well consider BA1 ,R1 instead of BA,R and the corresponding half-plane HA1 ,R1
instead of HA,R . Thus, there is no loss of generality in our assumption regarding
the multipliers M.
Now if γ ∈ k corresponds to M = [C, . . . , C]
∈ R and Q ∈ D corresponds
to the multiplier M, then for P ∈ BA,R and B = PP , we have
−1
B
Q
0
!
0
B = V −1 [[γ], . . . , [γ]]V,
Q
(141)
where [γ] = [γ(1) , . . . , γ(m) ]. Hence the eigenvalues γ1 , . . . , γn of Q are just
conjugates of γ and on the other hand, all conjugates of γ occur among the
eigenvalues of Q and Q. Moreover, from (??), we see readily that we can find
a non-singular matrix W uniformly for all Q ∈ D such that
W −1 QW = [γ1 , . . . , γn ].
(142)
We claim again that there is no loss of generality in assuming the complex
multiplications Q ∈ D to be already in the diagonal form as in (142). For,
we could have started with the Riemann matrix W −1 P instead of P right at the
beginning and then the corresponding D would have the desired property. We
could then have taken the corresponding half-plane HA,R .
We shall now study the effect of the involution in R on the multiplications
Q ∈ D. The involution in R corresponds to an automorphism γ → γ∗ in
k = Q(γ). This automorphism in k may be extended to all the conjugates k(i)
of k by prescription (γ(i) )∗ = (γ∗ )(i) . Thus, if M, M ∗ are the multipliers in R
corresponding to γ and γ∗ respectively, in k and Q, Q∗ are the corresponding
complex multiplications in D then Q = [γ1 , . . . , γn ] and Q∗ = [γ1∗ , . . . , γn∗ ]. On
the other hand we know from p. 173 that
′
−1
Q∗ = HQ H ,
i.e.
′
Q∗ H = HQ ,
181
(143)
H being positive hermitian. If h1 , . . . , hn are the diagonal elements of H, then
hi > 0 and moreover, from (??), we have
γk∗ hk = hk γk , k = 1, 2, . . . , n,
i.e.
γk∗ = γk , k = 1, 2, . . . , n.
Modular Functions and Algebraic Number Theory
161
Thus the automorphism in the conjugate fields k(i) is given by
γ(i) → (γ(i) )∗ = γ(i) .
(144)
We now contend that k is either totally real or totally complex. For, if k
has at least one real conjugate say k(1) , then from (144), we see that the ∗automorphism is identity on k(1) and hence on all conjugates of k. But then
γ(i) = (γ(i) )∗ = γ(i) .
i.e. all conjugates of k are real. Thus k is totally real and in the contrary case, k
is totally complex. In the latter case, we know that the ∗-automorphism is not
identity on k and has a fixed field k1 ∈ k such that k has degree 2 over k1 . But
k1 is necessarily totally real since the ∗-automorphism is identity on k1 and all
its conjugates. Thus, in the second case, k is an imaginary quadratic extension
of a totally real field k1 of degree m/2.
In the case when k is totally real, q = 2n/m is necessarily even. For, if M ∈
R corresponds to γ ∈ k and if Q ∈ D corresponds to M ∈ R, then the eigenvalues of Q are γ1 , . . . , γn and the eigenvalues of M are just γ1 , . . . , γn , γ1 , . . . , γn
i.e. γ1 , . . . , γn , γ1 , . . . , γn . But we know that the set γ1 , . . . , γn , γ1 , . . . , γn has
to run over the full set of conjugates γ(1) , . . . , γ(m) exactly q times. So necessarily that set γ1 , . . . , γn itself has to run over the complete set of conjugates
γ(1) , . . . , γ(m) a certain number of times, i.e. in other words, q is necessarily
even.
Thus when k is totally real, n = (q/2)m and when k is totally complex,
n = (m/2)q.
We are not interested in the latter case since, as it may be verified, the
space HA,R in this case, is of complex dimension zero and we can have no
useful function-theory in this space.
We shall therefore consider only the case when the ring of multipliers R
of the given n-rowed Riemann matrix P, gives a faithful representation of a
totally real algebraic number field k of degree m = n/(q/2) and moreover, we
may suppose that m is as large as possible i.e. q = 2 and m = n. The other
cases may be dealt with in a similar fashion.
We now prove the following
Theorem 15. Let P be an n-rowed Riemann matrix polarized with respect to a
rational alternate matrix A and let the ring of multipliers R of P give a faithful
representation of a totally real algebraic number field k of degree n over Q.
Then the generalized half-plane HA,R is the product of n complex half-planes
and the modular group ΓA,R is of the type of the Hilbert modular group.
182
Modular Functions and Algebraic Number Theory
162
Proof. (We shall retain, in the course of this proof, the simplification carried
out already regarding R and D and shall follow our earlier notation throughout).
We know that if k = Q(λ), then, for P ∈ BA,R ,
[λ]PV −1 = PV −1 [[λ][λ]],
(145)
where [λ] = [λ(1) , . . . , λ(n) ] and V = [Ω, Ω].
Let us write PV −1 as ((ri j )(si j )) where (ri j ) and (si j ) are n-rowed square
matrices. It follows from (145) then, that
ri j λ( j) = λ(i) ri j ,
si j λ( j) = λ(i) si j .
Since, for i , j, λ(i) , λ( j) we see that PV −1 is necessarily of the form ([ξ][η])
where [ξ] = [ξ1 , . . . , ξn ] and [η] = [η1 , . . . , ηn ].
Moreover, since M ∗ = M, we have MA = AM ′ for all M ∈ R. In particular,
we see that
[[λ], [λ]]V AV ′ = V AV ′ [[λ], [λ]].
K L with n-rowed square matrices K, L, M, N, we see
Splitting up V AV ′ and M
N
again, by the same argument as above, that
!
[µ] [ν]
′
V AV =
[κ] [ρ]
where [µ], [ν], [κ] and [ρ] and n-rowed diagonal matrices. Since A is alternate,
necessarily we have
!
0
[ν]
′
,
V AV =
−[ν] 0
where [ν] = [ν1 , . . . , νn ]. But now since A is rational, Ω−1 [ν]Ω′ −1 is rational.
As a consequence, Ω−1 [ν]Ω is again rational and hence by an earlier argument
(see p. 174), we have [ν] = [ν(1) , . . . , ν(n) ] for a ν ∈ k. Thus
!
0
Ω−1 [ν]Ω′ −1
A=
.
0
−Ω−1 [ν]Ω′ −1
It is to be noted that ν , 0 and further Ω−1 [ν]Ω′ −1 is not diagonal, in general.
Let now P ∈ BA,R be the representative of a point in HA,R . Then from the
fact that i−1 PAP′ > 0, we obtain
−1
i−1 PV V AV ′ (PV −1 )′ > 0,
183
Modular Functions and Algebraic Number Theory
i.e.
i−1 ([ξ][η])
163
!
0
[ν]
([ξ][η]) > 0,
−[ν] 0
i.e.
0 < hk = ν(k) i−1 (ξk ηk − ηk ξk ), for k = 1, 2, . . . , n.
If we set τ j = η j ξ−1
j , j = 1, 2, . . . , n and [τ] = [τ1 , . . . , τn ], then
[ξ]−1 PV −1 = (E (n) [τ])
(146)
and the conditions above are just
ν( j) Im(τ j ) > 0, j = 1, 2, . . . , n.
Now [ξ] commutes with all the multiplications Q ∈ Q since they are in the
diagonal form. Hence we could take [ξ]−1 P instead of P as a representative of
the corresponding point in HA,R . Thus, in view of (146), since V is independent
of P, the space HA,R may be identified as the set of n-tuples (z1 , . . . , zn ) with
complex zi satisfying ν(i) Im(zi ) > 0 i.e. HA,R is a product of n half-planes. If
ν > 0, then HA,R is just the product of n upper half-planes. For the general
theory, we should take HA,R as the product of n (not necessarily upper) halfplanes. The first statement in our theorem has thus been proved.
We shall now determine in explicit terms, the group ΓA,R which consists of
2n-rowed unimodular matrices U such that UAU ′ = A and U M = MU for all
M ∈ R.
Since U M = MU for all M ∈ R, we have, in particular,
UV −1 [[λ][λ]]V = V −1 [[λ][λ]]VU,
i.e.
VUV −1 [[λ][λ]] = [[λ][λ]]VUV −1 .
By the same argument as above, we have necessarily
!
[δ] [β]
VUV −1 =
,
[γ] [α]
where [α], [β], [γ], [δ] are diagonal matrices. Again, since the elements of
Ω−1 [δ]Ω are rational integers, it follows by an earlier remark (see p. 174)
that [δ] = [δ(1) , . . . , δ(n) ] for a δ ∈ k. Similarly, [β] = [β(1) , . . . , β(n) ], [γ] =
[γ(1) , . . . , γ(n) ] and [α] = [α(1) , . . . , α(n) ] for α, β, γ ∈ k. Let us denote by D the
order in k, consisting of numbers ρ ∈ k for which the matrix Ω−1 [ρ(1) , . . . , ρ(n) ]Ω
184
Modular Functions and Algebraic Number Theory
164
is a rationla integral matrix. Then α, β, γ, δ ∈ D. The order D coincides with
the ring o of all algebraic integers in k, when ω1 , . . . , ωn is a basis of o.
Now, from UAU ′ = A, we have
!
!
!
!
[δ] [β]
0
[ν] [δ] [γ]
0
[ν]
=
[γ] [α] −[ν] 0 [β] [α]
−[ν] 0
i.e.
185
ν(k) (α(k) δ(k) − β(k) γ(k) ) = ν(k) , k = 1, 2, . . . , n,
i.e.
αδ − βγ = 1.
Thus ΓA,R consists of 2n-rowed matrices of the form
!
[β]
−1 [δ]
V
V
[γ] [α]
where α, β, γ, δ are elements of D satisfying αδ − βγ = 1.
We know how ΓA,R acts on BA,R namely, if U ∈ ΓA,R , then U takes P ∈ BA,R
to PU. Now we may describe the action of ΓA,R on HA,R as follows. If the point
(τ1 , . . . , τn ) ∈ HA,R corresponds to the matrix PV −1 with P ∈ BA,R , then the
element U of ΓA,R maps (τ1 , . . . , τn ) on the point in HA,R which corresponds to
PUV −1 . But now
PUV −1 = PV −1 VUV −1
[δ]
= [ξ](E[τ])
[γ]
[β]
[α]
!
= [ξ]([δ] + [γ][τ][β] + [α][τ]).
Hence to PUV −1 corresponds the point
α(1) τ1 + β(1)
α(n) τn + β(n)
,
.
.
.
,
γ(1) τ1 + δ(1)
γ(n) τn + δ(n)
!
in HA,R . Thus ΓA,R is represented as the group of fractional linear transformations
!
α(1) τ1 + β(1)
α(n) τn + β(n)
(τ1 , . . . , τn ) → (1)
,
.
.
.
,
γ τ1 + δ(1)
γ(n) τn + δ(n)
(of HA,R onto itself) with α, β, δ ∈ D such that αδ − βγ = 1. If ω1 , . . . , ωn is
a basis of o, ΓA,R is the well-known Hilbert modular group. In other cases,
ΓA,R is seen to be commensurable with the Hilbert modular group.
Modular Functions and Algebraic Number Theory
165
Theorem 15 is therefore completely proved.
We have considered above only the case m = n. The case when n/m = p >
1 can be dealt with in a similar fashion. Here HA,R is a product of n generalized
half-planes of degree p and ΓA,R is the group of mappings
186
Z = (Z1 , . . . , Zn ) ∈ HA,R → (Z1∗ , . . . , Zn∗ ) ∈ HA,R
where Zi∗ = (A(i) Zi + B(i) )(C (i) Zi + D(i) )−1 and CA DB is a modular matrix of
degree p, with elements which are algebraic integers in k. The field of modular
functions belonging to ΓA,R in this case has been systematically investigated by
I.I. Piatetskii-Shapiro in recent years.
In the succeeding sections, we shall define the Hilbert modular function
and one of the main results which we wish to prove is that the Hilbert modular
functions form an algebraic function field of n variables. Our proof will run
essentially on the same lines as that of K.B. Gundlach. But first we need to
construct a workable fundamental domain in the space HA,R for the Hilbert
modular group. This shall be done in the next section.
2 Fundamental domain for the Hilbert modular
group
Let K be a totally real algebraic number field of degree n over Q, the field of
rational numbers and let K (1) (= K), K (2) , . . . , K (n) be the conjugates of K. For
b by adding an element ∞ which
our purposes in future, we first extend K to K
satisfies the following conditions, namely
a + ∞ = ∞, for a complex number a
a · ∞ = ∞, for complex a , 0
∞·∞=∞
1
=∞
0
1
= 0.
∞
(The expressions ∞ ± ∞, 0 · ∞ and ∞/∞ are undefined). We define the conjugates ∞(i) of ∞ to be ∞, again.
Corresponding to each K (i) , i = 1, 2, . . . , n, we associate a variable zi in
the extended complex plane subject to the restriction that if one zi is ∞, all the
b
other zi are also ∞. We shall denote the n-tuple (z1 , . . . , zn ) by z and for λ ∈ K,
we shall denote the n-tuple (λ(1) , . . . , λ(n) ) again by λ, when there is no risk of
187
Modular Functions and Algebraic Number Theory
166
confusion. For given z = (z1 , . . . , zn ) the norm N(z) shall stand for z1 , . . . , zn
and the trace S (z) for z1 + · · · + zn . If λ ∈ K, N(λ) and S (λ) coincide with the
usual norm and trace in K respectively.
Let S denote the group of 2-rowed square matrices S = αγ βδ with α, β,
γ, δ ∈ K and
αδ − βγ =
1. Let E denote the normal subgroup of S consist1
0
−1
0
ing of 0 1 and 0 −1 . Then, for the factor group S/E, we have a faithful
representation as the group of mappings
z = (z1 , . . . , zn ) → z M = (z∗1 , . . . , z∗n )
with
z∗j =
α( j) z j + β( j)
γ( j) z j + δ( j)
corresponding to each M = αγ βδ ∈ S. We shall denote z M as (αz + β)/(γz + δ)
symbolically. Let, further, z j = x j + iy j , z∗j = x∗j + iy∗j . Writing z = x + iy where
x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) we shall denote (y∗1 , . . . , y∗n ) by y M and
(x1∗ , . . . , xn∗ ) by x M so that z M = x M + iy M . It is easy to see that for M1 , M2 ∈ S,
z M1 M2 = (z M2 )M1 .
!
α(1)
α(n)
As usual, for M = γ δ ∈ S, ∞ M = (1) , . . . , (n) .
γ γ
Let M be the subgroup of S consisting of αγ βδ with α, β, δ ∈ o (the ring of
integers in K). The factor group M/E is precisely the (inhomogeneous) Hilbert
modular group, which we shall denote hereafeter by Γ.
One can
more generally than M, the group M0 of 2-rowed square
consider
matrices αγ βδ where α, β, γ, δ ∈ o and αδ − βγ = ǫ with ǫ being a totally
positive
unit
in K. Let E0 be the subgroup of M0 consisting of matrices of the
form 0ǫ 0ǫ where ǫ is a unit in K. Further, let ρ1 (= 1), ρ2 , . . . , ρν be a complete
set of representatives of the group of units ǫ > 0 in K modulo the subgroup of
squares of units in K. Then it is clear that M0 /E0 is isomorphic to the group
of substitutions z → ρz M where z → z M is a Hilbert modular substitution and
ρ = ρi for some i. Thus the study of M0 /E0 can be reduced to that of Γ. The
group Γ is, in general, “smaller than” M0 /E0 and is called, therefore the narrow
Hilbert modular group, usually.
b are equivalent (in symbols, λ ∼ µ), if for some
Two
λ, µ in K
elements
M = αγ βδ ∈ M,
αλ + β
µ = λM =
.
γλ + δ
This is a genuine equivalence relation. We shall presently show that K falls
into h equivalence classes, where h is the class number of K.
α β
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Modular Functions and Algebraic Number Theory
167
b such that for any λ ∈ K,
b we have
Proposition 20. There exist λ1 , . . . , λh ∈ K
λ ∼ λi for some i uniquely determined by λ.
b is of the form ρ/σ where ρ, σ ∈ o. (If λ = ∞, we may
Proof. Any λ ∈ K
take ρ = 1 and σ = 0). With λ, we may associate the integral ideal a = (ρ, σ).
We first see that if a1 = (θ)a is any integral ideal in the class of a, then a1 is
of the form (ρ1 , σ1 ) where ρ1 and σ1 are in o such that λ = ρ1 /σ1 . This is
quite obvious, since a1 = (ρθ, σθ) and taking ρ1 = ρθ, σ1 = σθ, our assertion
b in this way
is proved. Conversely, any integral ideal a1 associated with λ ∈ K
is necessarily in the same ideal-class as a. In fact, let λ be written in the form
ρ1 /σ1 with ρ1 , σ1 ∈ o and let a1 = (ρ1 , σ1 ). Then we claim a1 is in the same
class as a. For, since λ = ρ/σ = ρ1 /σ1 , we have ρσ1 = ρ1 σ and hence
(ρ) (σ1 ) (ρ1 ) (σ)
=
a a1
a1 a
Now (ρ)/a and (σ)/a as also (ρ1 )/a1 and (σ1 )/a1 are mutually coprime. Hence
we have (ρ)/a = (ρ1 )/a1 and similarly (σ)/a = (σ1 )/a1 . This means that for
aθ ∈ K, ρ1 = ρθ and hence σ1 = σθ. Thus a1 = (θ)a, for θ ∈ K.
We choose in the h ideal classes, fixed integral ideals a1 , . . . , ah such that
ai is of minimum norm among all the integral ideals of its class. (Perhaps,
the ideal ai is not uniquely fixed in its class by this condition but there are at
most finitely many possibilities for ai and we choose from these, a fixed ai ).
b we can make correspond an
It follows from the above that to a given λ ∈ K,
ideal ai such that λ = ρ/σ and ai = (ρ, σ) for suitable ρ, σ ∈ o. Now, if
αλ + β
∼ λ, then to µ again corresponds the ideal (αρ + βσ, γρ + δσ)
µ =
γλ + δ
which is just ai since αγ βδ is unimodular. Thus all elements of an equivalence
b correspond to the same ideal.
class in K
b then
We shall now show that if the same ideal ai corresponds to λ, λ∗ ∈ H,
∗
∗
∗
∗
necessarily λ ∼ λ . Let, in fact, λ = ρ/σ, λ = ρ /σ and ai = (ρ, σ) = (ρ∗ , σ∗ ).
It is well-known that there exist numbers
ξ, η, ξ∗ , η∗ in a−1
i such that ρη−σξ = 1
ρ ξ
∗ ∗
∗ ∗
and ρ η − σ ξ = 1. Let A = σ η and
ρ∗
A = ∗
σ
∗
then A, A∗ ∈ S and moreover
A∗ A−1 =
ρ∗
σ∗
!
ξ∗ η
η∗ −σ
!
ξ∗
;
η∗
!
!
α β
−ξ
=
γ δ
ρ
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Modular Functions and Algebraic Number Theory
168
where
α = ρ∗ η − ξ∗ σ, β = −ρ∗ ξ + ξ∗ ρ, γ = σ∗ η − η∗ σ, δ = −σ∗ ξ + η∗ ρ
clearly are integers in K satisfying αδ − βγ = 1. Hence A∗ A−1 ∈ M and thus
λ∗ =
ρ∗
αρ + βσ αλ + β
=
=
∼ λ.
∗
σ
γρ + δσ
γλ + δ
b correspond different
Thus, we see that to different equivalence classes in H
ideals ai . It is almost trivial to verify that to each ideal ai , there corresponds
b As a consequence, there are exactly h equivalence
an equivalence class in K.
b
classes in K and our proposition is proved.
We now make a convention to be followed in the sequel. We shall assume
that λ1 = 1/0 = ∞, without loss of
Moreover, with λi = ρi /σi ,
generality.
we associated a fixed matrix, Ai = σρii ηξii ∈ S; let us remark that it is always
−1
possible to find, though not uniquely, numbers
ξi , ηi in ai such that ρi ηi −ξi σi =
1
0
1. Further, we shall suppose that A1 = 0 1 .
b Γλ denote the group of Hilbert modular substitutions z →
Let, for λ ∈ K,
(αz + β)/(γz + δ) such that (αλ + β)/(γλ + δ) = λ. For our use later, we need to
determine Γλ explicitly. It clearly suffices to find Γλi (for i = 1, 2, . . . , h) since
λ = (λi ) M for some M ∈ Γ and Γλ = MΓλi M −1 . Let then for
!
αλi + β
α β
= λi .
M=
∈ M,
γ δ
γλi + δ
We have (αρi + βσi , γρi + δσi ) = ai = (ρi , σi ). And since (αρi + βσi )σi =
(γρi + δσi )ρi , we have
(αρi + βσi ) (σi ) (γρi + δσi ) (ρi )
=
.
ai
ai
ai
ai
As before,
(αρi + βσi ) (ρi )
=
ai
ai
and
(γρi + δσi ) (σi )
=
.
ai
ai
Hence, for a unit ǫ in K, we have
αρi + βσi = ǫρi , γρi + δσi = ǫσi .
This means that
MAi =
ρi
σi
!
!
ξi∗ ǫ 0
,
η∗i 0 ǫ −1
190
Modular Functions and Algebraic Number Theory
169
where ξi∗ = (αξi +βηi )ǫ and η∗i = (γξi +δηi )ǫ lie in a−1
i . Further since ρi ηi −σi ξi =
1 = ρi η∗i − σi ξi∗ , we have
ρi (η∗i − ηi ) = σi (ξi∗ − ξi ),
i.e.
(ρi ) (η∗i − ηi ) (σi ) (ξi∗ − ξi )
=
.
ai
ai
a−1
a−1
i
i
Again, since (ρi )/ai is coprime to (σi )/ai , we see that (ρi )/ai divides
i.e. (ξi∗ − ξi ) = a−2
i i(ρi ) for an integral ideal i. In other words,
(ξi∗ − ξi )
a−1
i
,
ξi∗ = ξi + ρi ζ, ζ ∈ a−2
i .
As a result, we obtain also
η∗i = ηi + σi ζ.
We now observe that
!
ǫ ζǫ −1
.
MAi = Ai
0 ǫ −1
Thus Γλi consists
precisely of the modular substitutions z → z M , where
M = Ai 0ǫ ǫζ−1 A−1
∈
M
with ζ ∈ a−2
i
i and ǫ any unit in K.
Let Hn denote the product of n upper half-planes, namely, the set of z =
(z1 , . . . , zn ) with z j = x j + iy j , y j > 0. The Hilbert modular group Γ has a
representation as a group of analytic homeomorphisms z → (αz
+ δ)
+ β)/(γz
of Hn onto itself. In the following, we shall freely identify M = αγ βδ ∈ M with
the modular substitution z → z M in Γ and speak of M belonging to Γ, without
risk of confusion.
b lie on the boundary of Hn and
The points λ = (λ(1) , . . . , λ(n) ) for λ ∈ K
are called (parabolic) cusps of Hn . By Proposition 20, there exist h cusps,
(n)
(1)
(n)
λ1 = (∞, . . . , ∞), λ2 = (λ(1)
2 , . . . , λ2 ), . . . , λh = (λh , . . . , λh ) which are not
equivalent with respect to Γ and such that any other (parabolic) cusp of Hn is
equivalent to exactly one of them; λ1 , . . . , λh are called the base cusps.
If V ⊂ C n , then for given M ∈ Γ, V M shall denote the set of all z M for z ∈ V.
For any z ∈ Hn , Γz shall stand for the isotropy group of z in Γ, namely the group
of M ∈ Γ, for which z M = z. We shall conclude from the following proposition
that Γz is finite, for all z ∈ Hn .
Proposition 21. For any two compact sets B, B′ in Hn , the number of M ∈ Γ
for which BM intersects B′ is finite.
191
Modular Functions and Algebraic Number Theory
170
Proof. Let Λ denote the set of M = αγ βδ ∈ Γ such that BM intersects B′
i.e. given M ∈ Λ, there exists w = (w1 , . . . , wn ) in B such that w M = (w′1 , . . . , w′n )
∈ B′ . Let w j = u j + iv j and w′j = u′j + iv′j ; then v′j = v j |γ( j) w j + δ( j) |−2 . Since B
and B′ are compact, we see that
192
|γ( j) w j + δ( j) | ≤ c, j = 1, 2, . . . , n
for a constant c depending only on B and B′ . But then itfollows
immediately
0 1 and let B∗ and
that γ, δ belong to a finite set of integers in K. Let I = −1
0
B′ ∗ denote the images of B and B′ respectively under the modular substitution
Further noting
z → zI = −z−1 . The images B∗ and B′ ∗ are again
δ compact.
−γ
,
we
can
show easily
that if M = αγ βδ ∈ Γ, then I MI ′ = M ′ −1 = −β
α
that BM ∩ B′ , ∅ if and only if B∗M′ −1 ∩ B′ ∗ , ∅. Now applying the same
∗
′∗
argument
α β as above to the compact sets B and B , we may conclude that if, for
∗ ′
′∗
M = γ δ ∈ Γ, (B ) M −1 ∩ B , ∅, then α, β belong to a finite set of integers
in K. As a result, we obtain finally that Λ is finite.
Taking a point z ∈ Hn instead of B and B′ , we deduce that Γz is finite.
We are now in a position to prove the following
Theorem 16. The Hilbert modular group Γ acts properly discontinuously on
Hn ; in other words, for any z ∈ Hn , there exists a neighbourhood V of z such
that only for finitely many M ∈ Γ, V M intersects V and when V M ∩ V , ∅, then
M ∈ Γz .
Proof. Let z ∈ Hn and V a neighbourhood of z such that the closure V of V
in Hn is compact. Taking V for B and B′ in Proposition 21, we note that only
for finitely many M ∈ Γ, say M1 , . . . , Mr , V M intersects V. Among these Mi ,
let M1 , . . . , M s be exactly those which do not belong to Γz . Then we can find a
neighbourhood W of z such that W Mi ∩W = ∅, i = 1, 2, . . . , s. Now let U = W ∩
V; then U has already the property that for M , Mi (i = 1, 2, . . . , r)U M ∩U = ∅.
Further U Mi ∩ U = ∅, for i = 1, 2, . . . , s. Thus U satisfies the requirements of
the theorem.
If z is not a fixed point of any M in r except the identity of in other words,
if Γz consists only of 10 01 , then there exists a neighbourhood V z such that for
M , 10 01 , V M ∩ V = ∅.
Let λ = ρ/σ be a cusp of Hn and a = (ρ, σ) be the integral ideal among
−2
a1 , . . . , ah which is associated with λ. Let α1 , . . . , αn be a minimal
basis for a
ρ ξ
and further, associated with λ, let us choose a fixed A = σ η ∈ S with ξ, η
lying in a−1 . Moreover, let ǫ1 , . . . , ǫn−1 be n − 1 independent generators of the
193
Modular Functions and Algebraic Number Theory
171
group of units in K. As a first step towards constructing a fundamental domain
for Γ in Hn , we shall introduce ‘local coordinates’ relative to λ, for every point
z in Hn and then construct a fundamental domain Gλ , for Γλ in Hn .
Let z = (z1 , . . . , zn ) be any point of Hn and zA−1 = (z∗1 , . . . , z∗n ) with z∗j =
∗
x j + iy∗j . Denoting yA−1 = (y∗1 , . . . , y∗n ) by y∗ , we define the local coordinates of
z relative to λ by the 2n quantities
p
1/ N(y∗ ), Y1 , . . . , Yn−1 , X1 , . . . , Xn ,
where Y1 , . . . , Yn−1 , X1 , . . . , Xn are uniquely determined by the linear equations

y∗
1

(k)

Y1 log |ǫ1(k) | + · · · + Yn−1 log |ǫn−1
| = log pn k , 


∗
2

N(y ) 


(147)


k = 1, 2, . . . , n − 1 





X α(1) + · · · + X α(1) = x∗ , l = 1, 2, . . . , n.
1 1
n n
l
The group Γλ consists of allthe modular
substitutions of the form z → z M
ǫζǫ −1
where M = AT A−1 and T = 0ζǫ
with
ǫ,
a unit in K and ζ ∈ a−2 . The
−1
transformation z → z M is equivalent to the transformation zA−1 → zT A−1 =
ǫ 2 zA−1 + ζ. It is easily verified that Γλ is generated by the ‘dilatations’ zA−1 →
ǫi2 zA−1 , i = 1, 2, . . . , n − 1 and the ‘translations’ zA−1 → zA−1 + α j , j = 1, 2, . . . , n.
ǫζǫ −1 Let now z → z M with M = AT A−1 , T = 0ζǫ
be a modular substi−1
kn−1
tution in Γλ and let ǫ = ±ǫ1k1 . . . ǫn−1
and ζ = m1 α1 + · · · + mn αn , where
k1 , . . . , kn−1 ,pm1 , . . . , mn are rational integers. It is obvious that the first coordinate 1/ N(y∗ ) of z is preserved by the modular substitution z → z M since
N(yA−1 M ) = N(yT A−1 ) = N(ǫ 2 )N(yA−1 ) = N(y∗ ). The effect of the substitution
z → z M on Y1 , . . . , Yn−1 is given by
(Y1 , Y2 , . . . , Yn−1 ) → (Y1 + k1 , Y2 + k2 , . . . , Yn−1 + kn−1 ),
(148)
as can be verified from (147). If ǫ 2 = 1, then the effect of the substitution
z → z M on X1 , . . . , Xn is again just a ‘translation’,
(X1 , . . . , Xn ) → (X1 + m1 , . . . , Xn + mn ).
(149)
If ǫ 2 , 1, then the effect of the substitution z → z M on X1 , . . . , Xn is not merely
a ‘translation’ but an affine transformation given by
(X1 , . . . , Xn ) → (X1∗ + m1 , . . . , Xn∗ + mn ),
where (X1∗ X2∗ . . . Xn∗ ) = (X1 X2 . . . Xn )UVU −1 , U denotes the n-rowed square
2
2
matrix (α(i j) ) and V is the diagonal matrix [ǫ (1) , . . . , ǫ (n) ].
194
Modular Functions and Algebraic Number Theory
We define a point z ∈ Hn to be reduced with respect to Γλ , if

1
1


− ≤ Yi < , i = 1, 2, . . . , n − 1,



2
2



1
1



− ≤ X j < , j = 1, 2, . . . , n.
2
2
172
(150)
It is, first of all, clear that for any z ∈ Hn , there exists an M ∈ Γλ such that the
kn−1
equivalent point z M is reduced with respect to Γλ . In fact, for ǫ = ±ǫ1k1 . . . ǫn−1
and M1 = A 0ǫ ǫ0−1 A−1 , the effect of the substitution z → z M1 is given by
(??) and hence, by choosing k1 , . . . , kn−1 properly, we could suppose that the
coordinates Y1 , . . . , Yn−1 of
z M 1 satisfy (150). Again, since for ζ = m1 α1 +
· · · + mn αn and M2 = A 10 1ζ A−1 , the effect of the substitution z → z M2 on
the coordinates X1 , . . . , Xn of z M1 is given by (149), we could suppose that for
suitable m1 , . . . , mn , the coordinates X1 , . . . , Xn of z M2 ,M1 satisfy (150). It is now
clear that z M2 M1 is reduced with respect to Γλ . On the other hand, let z = x + iy,
w = u + iv ∈ Hn be reduced and equivalent with respect to Γλ and let the local
coordinates of z and w relative to λ be respectively
p
1/ N(yA−1 ), Y1 , . . . , Yn−1 , X1 , . . . , Xn ,
p
∗
1/ N(vA−1 ), Y1∗ , . . . , Yn−1
, X1∗ , . . . , Xn∗ .
Further, let zA−1 = ǫ 2 wA−1 + ζ. Then, in view of the fact that
Yi∗ ≡ Yi (
mod 1), −
195
1
1
≤ Yi , Yi∗ < , i = 1, 2, . . . , n − 1,
2
2
we have first Yi∗ = Yi , i = 1, 2, . . . , n − 1 and hence ǫ 2 = 1. Again, since we
have
1
1
X ∗j ≡ X j ( mod 1), − ≤ X j , X ∗j < , j = 1, 2, . . . , n,
2
2
we see that X ∗j = X j , j = 1, 2, . . . , n and hence ζ = 0. Thus zA−1 = wA−1 ,
i.e. z = w.
Denoting by Gλ the set of z ∈ Hn whose local coordinates Y1 , . . . , Yn−1 , X1 , . . . , Xn
satisfy (150), we conclude that any z ∈ Hn is equivalent with respect to Γλ , to a
point of Gλ and no two distinct points of Gλ are equivalent with respect to Γλ .
Thus Gλ is a fundamental domain for Γλ in hn .
For n = 1, the fundamental domain Gλ is just the vertical strip −1/2 ≤ x∗ <
1/2, y∗ > 0 with reference to the coordinate zA−1 = x∗ + iy∗ . Going back to the
coordinate z, the vertical strips x∗ = ±1/2, y∗ > 0 are mapped into semi-circles
passing through λ and orthogonal to the real axis and the fundamental domain
looks as in the figure.
Modular Functions and Algebraic Number Theory
173
Figure, page 195
If z ∈ Gλ and if c1 ≤ N(yA−1 ) ≤ c2 , then we claim that z lies in a compact
set in Hn depending on c1 , c2 and on the choice of ǫ1 , . . . , ǫn−1 and α1 , . . . , αn
in K. As a matter of fact, from (150), we know that


 y∗i 
 ≤ c3 for i = 1, 2, . . . , n − 1,
log  p
n
N(y∗ ) and |x∗j | ≤ c4 for j = 1, 2, . . . , n. Hence we have
c5 ≤ pn
y∗i
N(y∗ )
≤ c6 ,
for i = 1, 2, . . . , n − 1.
Since c1 ≤ N(y∗ ) ≤ c2 , we have indeed for all i = 1, 2, . . . , n,
c7 ≤ pn
y∗i
N(y∗ )
≤ c8 .
As a result, we obtain for i, j = 1, 2, . . . , n,
c9 ≤ y∗i ≤ c10 ,
|x∗j | ≤ c4 ,
where c9 , c10 and c4 depend only on c1 , c2 and the choice of ǫ1 , . . . , ǫn−1 ,
α1 , . . . , αn in K. Thus zA−1 and therefore z lies in a compact set in Hn , depending on c1 , c2 and K.
Now, we introduce the notion of “distance of a point z ∈ Hn from a cusp λ
of Hn ”. We have already in Hn a metric given by
ds2 =
n
X
dx2 + dy2
i
i=1
i
y2i
which is non-euclidean in the case n = 1 and has an invariance property with
respect to Γ. But since the cusps lie on the boundary of Hn , the distance (relative
to this metric) of an inner point of Hn from a cusp is infinite. Hence, this metric
is not useful for our purposes.
For z ∈ Hn and a cusp λ = ρ/σ with associated A = σρ ηξ ∈ S, we define
the distance ∆(z, λ) of z from λ by
1
1
∆(z, λ) = (N(yA−1 ))− 2 = (N(y−1 | − σz + ρ|2 )) 2
196
Modular Functions and Algebraic Number Theory
174
1
= (N((−σx + ρ)2 y−1 + σ2 y)) 2 .
p
For λ = ∞, ∆(z, ∞) = 1/ N(y); hence the larger the N(y), the ‘closer’ is z to
∞.
Now ∆(z, λ) has an important invariance property with respect to Γ. Namely,
for M ∈ Γ, we have
∆(z M , λ M ) = ∆(z, λ).
(151)
This is very easy to verify, since, by definition
197
1
∆(z M , λ M ) = (N(Im(z M )A−1 M−1 ))− 2
1
= (N(yA−1 ))− 2 = ∆(z, λ).
Moreover, ∆(z, λ) doesnot depend on the special choice of A associated
with λ. For, if A1 = ∼ρ1i ηξ11 ∈ S is associated with λ = ρ1 /σ1 , then A−1 A1 =
T = 0ǫ ǫζ−1 where ǫ is a unit in K. Now N(yA−1
) = N(yA−1 )N(ǫ −2 ) = N(yA−1 )
1
and hence our assertion is proved.
Let, for a given cusp λ and r > 0, Uλ,r denote the set of z ∈ Hn such
that ∆(z, λ) < r. This defines a ‘neighbourhood’ of λ and all points z ∈ Hn
which belong to Uλ,r are inner points of the same. The neighbourhoods, Uλ,r
for 0 < r < ∞ cover the entire upper half-plane Hn . Each neighbourhood Uλ,r
is left invariant by a modular substitution in Γλ . For, by (151), if M ∈ Γλ , then
∆(z M , λ) = ∆(z M , λ M ) = ∆(z, λ)
and so, if z ∈ Uλ,r , then again ∆(z M , λ) < r i.e. z M ∈ Uλ,r . A fundamental
domain for Γλ in aλ,r is obviously given by Gλ ∩ Uλ,r .
We shall now prove some interesting facts concerning ∆(z, λ) which will be
useful in constructing a fundamental domain for Γ in Hn .
(i) For z = x + iy ∈ Hn , there exists a cusp λ of Hn such that for all cusps µ
of Hn , we have
∆(z, λ) ≤ ∆(z, µ).
(152)
Proof. If λ is a cusp, then λ = ρ/σ for ρ, σ ∈ o such that (ρ, σ) is one of the h
ideals U1 , . . . , Uh . Then
1
∆(z, λ) = (N((−σx + ρ)2 y−1 + σ2 y)) 2 .
Modular Functions and Algebraic Number Theory
175
1
Let us consider the expression (N((−σx + ρ)2 y−1 + σ2 y)) 2 as a function of
the pair of integers (ρ, σ). It remains unchanged if ρ, σ are replaced by ρǫ, σǫ
for any unit ǫ in K. We shall now show that there exists a pair of integers ρ1 ,
σ1 in K such that
1
1
(N((−σ1 x + ρ1 )2 y−1 + σ21 y)) 2 ≤ (N((−σx + ρ)2 y−1 + σ2 y)) 2
(153)
for all pairs of integers (ρ, σ). In order to prove (153), it obviously suffices to
show that for given c1 > 0, there are only finitely many non-associated pairs of
integers (ρ, σ) such that
1
(N((−σx + ρ)2 y−1 + σ2 y)) 2 ≤ c11 .
(154)
Now, it is known from the theory of algebraic number fields that if α =
(α1 , . . . , αn ) is an n-tuple of real numbers with N(α) , 0, then we can find a
unit ǫ in K such that
|αi ǫ (i) | ≤ c12 |N(α)|1/n
(155)
for a constant c12 depending only on K. In view of (154), we can suppose after
multiplying ρ and σ by a suitable unit ǫ, that already we have
2
(i)
(−σ(i) xi + ρ(i) )2 y−1
i + σyi ≤ c13 ,
i = 1, 2, . . . , n,
for a constant c13 depending only on c11 and c12 . This implies that (−σ(i) xi +ρ(i) )
and σ(i) and as a consequence ρ(i) and σ(i) are bounded, for i = 1, 2, . . . , n.
Again, in this case, we know from the theory of algebraic numbers that there
are only finitely many possibilities for ρ and σ. Hence (154) is true only for
finitely many non-associated pairs of integers (ρ, σ). From these pairs, we
1
choose a pair (ρ1 , σ1 ) such that the value of N((−σ1 x + ρ1 )2 y−1 + σ21 y)) 2 is a
minimum. This pair (ρ1 , σ1 ) now obviously satisfies (153).
Let (ρ1 , σ1 ) = b and let b = ai (θ)−1 for some ai and a θ ∈ K. Then ai =
(ρ1 θ, σ1 θ). Now, in (153), if we replace ρ, σ by ρ1 θ, σ1 θ respectively, we get
|N(θ)| ≥ 1. On the other hand, since ai is of minimum norm among the integral
ideals of its class, N(ai ) ≤ N(b) and therefore |N(θ)| ≤ 1. Thus |N(θ)| = 1.
Let now ρ = ρ1 θ, σ = σ1 θ and λ = ρ/σ. Then ai = (ρ, σ) and by definition,
1
1
∆(z, λ) = (N((−σx + ρ)2 y−1 + σ2 y)) 2 = (N((−σ1 x + ρ1 )2 y−1 + σ21 y)) 2 . If we
use (153), then we see at once that ∆(z, λ) ≤ ∆(z, µ) for all cusps µ.
For given z ∈ Hn , we define
Z
∆(z) =
∆(z, λ).
λ
198
Modular Functions and Algebraic Number Theory
176
By (i), there exists a cusp λ such that ∆(z, λ) = ∆(z). In general, λ is unique, but
there are exceptional cases when the minimum is attained for more than one λ.
We shall see presently that there exists a constant d > 0, depending only on K
such that if ∆(z) < d, then the cusp λ for which ∆(z, λ) = ∆(z) is unique.
199
(ii) There exists d > 0 depending only on K, such that, if for z = x + iy ∈ Hn ,
∆(z, λ) < d and ∆(z, µ) < d, then necessarily λ = µ.
Proof. Let λ = ρ/σ and µ = ρ1 /σ1 and let for a real number d > 0,
1
∆(z, λ) = (N((−σx + ρ)2 y−1 + σ2 y)) 2 < d,
1
∆(z, µ) = (N(−σ1 x + ρ1 )2 y−1 + σ21 y) 2 < d.
After multiplying ρ and σ by a suitable unit ǫ in K, we might assume in view
of (155) that
2
(i)
2/n
(σ(i) xi − ρ(i) )2 y−1
,
i + σ yi < c12 d
i = 1, 2, . . . , n.
Hence
√
c12 d1/n ,
1
√
|σ(i) |yi2 < c12 d1/n .
− 21
| − σ(i) xi + ρ(i) |yi
<
Similarly we have
√
c12 d1/n ,
1
√
2
|σ(i)
c12 d1/n .
1 |yi <
− 21
(i)
| − σ(i)
1 xi + ρ1 |yi
<
Now
−1
1
−1
1
(i) (i)
(i) 2
(i)
(i)
(i)
(i)
(i) 2
2
2
ρ(i) σ(i)
1 − ρ1 σ = (−σ xi + ρ )yi σ1 yi − (−σ1 xi + ρ1 )yi σ yi
and hence
|N(ρσ1 − σρ1 )| < (2c12 d2/n )n .
If we set d = (2c12 )−n/2 , then |N(ρσ1 −σρ1 )| < 1. Since ρσ1 −σρ1 is an integer,
it follows that ρσ1 − σρ1 = 0 i.e. λ = µ.
Thus for d = (2c12 )−n/2 , the conditions ∆(z, λ) < d, ∆(z, µ) < d for a z ∈ Hn
imply that λ = µ. Therefore, the neighbourhoods Uλ,d for the various cusps λ
are mutually disjoint.
We shall now prove that for z ∈ Hn , ∆(z) is uniformly bounded in Hn . To
this end, it suffices to prove
200
Modular Functions and Algebraic Number Theory
177
(iii) There exists c > 0 depending only on K such that for any z = x + iy ∈ Hn ,
there exists a cusp λ with the property that ∆(z, λ) < c.
Proof. We shall prove the existence of a constant c > 0 depending only on K
and a pair of integers (ρ, σ) not both zero such that
1
(N((−σx + ρ)2 y−1 + σ2 y)) 2 < c.
Let ω1 , . . . , ωn be a basis of o. Consider now the following system of 2n
linear inequalities in the 2n variables a1 , . . . , an , b1 , . . . , bn , viz.
1
y− 2 (ω(k) a + · · · + ω(k) a ) − x y− 21 (ω(k) b + · · · + ω(k) b ) ≤ α
k k
k
n n
n n 1 1
1 1
k
for k = 1, 2, . . . , n and
for k = 1, . . . , n.
1
y 2 (ω(k) b + · · · + ω(k) b ) ≤ β
k
n n k 1 1
|(ω(i j) )|2
The determinant of this system of linear forms is
= |∆| where ∆ is
the discriminant of K. By Minkowski’s theorem on linear forms, this system
of linear inequalities has a nontrivial solution in rational√integers a1 , . . . , an ,
b1 , . . . , bn if α1 , . . . , αn ·β1 , . . . , βn ≤ |∆|. Taking αi = β j = 2n |∆|i = 1, 2, . . . , n, j =
1, 2, . . . , n, in particular, this system of inequalities has a nontrivial solution in
rational integers, say, a′1 , . . . , a′n , b′1 , . . . , b′n . Let us take ρ = a′1 ω1 + · · · + a′n ωn
and σ = b′1 ω1 + · · · + b′n ωn . Then we obtain
p
(−σ(i) xi + ρ(i) )2 y−1 + σ(i)2 yi ≤ 2n |∆|, i = 1, 2, . . . , n.
i
1
1
Hence (N((−σx + ρ)2 y−1 + σ2 y)) 2 ≤ 2n/2 |∆| 2 = c (say).
Now let (ρ, σ) = b; b = ai (θ)−1 for some ai and for some θ ∈ K. Since ai
is of minimum norm among the integral ideals of its class, |N(θ)| ≤ 1. Further
ai = (ρθ, σθ). Now, for the cusp λ = ρθ/σθ = ρ/σ, we have
1
∆(z, λ) = |N(θ)|(N((−σx + ρ)2 y−1 + σ2 y)) 2 ≤ c,
which was what we wished to prove.
S
From (iii), we deduce that Hn = Uλ,c .
λ
(iv) For z ∈ Hn and M ∈ Γ, ∆(z M ) = ∆(z).
201
Modular Functions and Algebraic Number Theory
178
Proof. In fact,
∆(z M ) = inf ∆(z M , λ)
λ
= inf ∆(z, λ M−1 )
λ
((by 151))
= inf ∆(z, λ)
λ
= ∆(z).
We now have all the necessary material for the construction of a fundamental domain for Γ in Hn .
A point z ∈ Hn is semi-reduced (with respect to a cups λ), if ∆(z) = ∆(z, λ).
If z is semi-reduced with respect to λ, then for all cusps µ, we have ∆(z, µ) ≥
∆(z, λ).
Let λ1 (= (∞, . . . , ∞)), λ2 , . . . , λh be the h inequivalent base cusps of Hn .
We denote by Fλi , the set of all z ∈ Hn which are semi-reduced with respect to
λi . Clearly Fλi ⊂ Uiλi,c , in view of (iii) above. The set Fλi is invariant under the
modular substitution z → z M , for M ∈ Γλi . For, by (iv) above, ∆(z M ) = ∆(z)
and further.
∆(z) = ∆(z, λi ) = ∆(z M , (λi ) M ) = ∆(z M , λi ).
Thus ∆(z M ) = ∆(z M , λi ) and hence z M ∈ Fλi , for M ∈ Γλi .
Let Gλi denote the closure in Hn of the set Gλi and qi = Fλi ∩ Gλi . Then
qi is explicitly defined as the set of z ∈ Hn whose local coordinates X1 , . . . , Xn ,
Y1 , . . . , Yn−1 relative to the cusp λi satisfy the conditions
−
1 1
1
1
≤ Xk ≤ , − ≤ Y1 ≤ (k = 1, 2, . . . , n; l = 1, 2, . . . , n − 1)
2
2 2
2
and further, for all cusps µ,
∆(z, µ) ≥ ∆(z, λi ).
Let F = ∪hi=1 qi . We see that the qi as also F are closed in Hn .
A point z ∈ qi is an inner point of qi if in all the conditions above, strict
inequality holds, viz.
−
1
1 1
1
< Xk < , − < Yl < (k = 1, 2, . . . , n, l = 1, 2, . . . , n − 1)
2
2 2
2
∆(z, µ) > ∆(z, λi ), for µ , λi .
(150)∗
202
Modular Functions and Algebraic Number Theory
179
If equality holds even in one of these conditions, then z is said to be a boundary
point of qi . The set of boundary points of qi constitute the boundary of qi , which
may be denoted by Bd qi . We denote by Ri , the set of inner points of qi .
It is clear that the qi do not overlap and intersect at most on their boundary.
A point z ∈ F may now be called an inner point of F, if z is an inner point
of some qi ; similarly we may define a boundary point of F and denote the set
of boundary points of F by Bd F.
We say that a point z ∈ Hn is reduced (with respect to Γ) if, in the first place,
z is semi-reduced with respect to some one of the h cusps λ1 , . . . , λh , say λi and
then further z ∈ Gλi . Clearly F is just the set of all z ∈ Hn reduced with respect
to Γ.
Before we proceed to show that F is a fundamental domain for Γ in Hn , we
shall prove the following result concerning the inner points of F, namely,
The set of inner points of F is open in Hn .
Proof. It is enough to show that each R j is open in Hn . Let, then, z0 = x0 +iy0 ∈
R j ; we have to prove that there exists a neighbourhood V of z0 in Hn which is
wholly contained in R j .
Now, for each z = x + iy ∈ Hn and a cusp µ = ρ/σ, we have ∆(z, µ) =
1
(N((−σx + ρ)2 y−1 + σ2 y)) 2 . Using the fact that for each j = 1, 2, . . . , n,
2
(−σ( j) x j + ρ( j) )2 y−1 + σ( j) y j is a positive-definite binary quadratic form in σ( j)
and ρ( j) , we can easily show that
1
∆(z, µ) ≥ α(N(σ2 + ρ2 )) 2
where α = α(z) depends continuously on z and does not depend on µ. We can
now find a sufficiently small neighbourhood W of z0 such that for all z ∈ W,
1
1
α0 (N(σ2 + ρ2 )) 2
2
where α0 = α(z0 ). Moreover, we can assume W so chosen that for all z ∈ W,
∆(z, µ) ≥
∆(z, λ j ) ≤ 2∆(z0 , λ j ).
Thus, for all cusps µ = ρ/σ and z ∈ W,
1
1
α0 (N(σ2 + ρ2 )) 2 − 2∆(z0 , λ j ).
2
Now, employing an argument used already on p. 198, we can show that
there are only finitely many non-associated pairs of integers (ρ, σ) such that
∆(z, µ) − ∆(z, λ j ) ≥
1
1
α0 (N(σ2 + ρ2 )) 2 ≤ 2∆(z0 , λ j ).
2
203
Modular Functions and Algebraic Number Theory
180
It is an immediate consequence that except for finitely many cusps µ1 , . . . , µr ,
we have for µ , λ j ,
∆(z, µ) > ∆(z, λ j ).
Now, since ∆(z0 , µ) > ∆(z0 , λ j ) for all cusps µ , λ j , we can, in view of the
continuity in z of ∆(z, µi ) (for i = 1, 2, . . . , r) find a neighbourhood U of z0
such that
∆(z, µk ) > ∆(z, λ j ) (k = 1, 2, . . . , r)
for z ∈ U and µk , λ j . We then have finally for all z ∈ U ∩ W
∆(z, µ) > ∆(z, λ j )
provided µ , λ j . Further, we could have chosen U such that for all z ∈ U,
(150)∗ is satisfied, in addition. Thus the neighbourhood V = U ∩ W of z0
satisfies our requirements and so R j is open.
It may now be seen that the closure Ri of Ri is just qi . In fact, let z ∈ qi and
= ν = ρ/σ , ∞. Then we have
, µA−1
let λi = ∞Ai , z∗ = zA−1
i
i
σ , 0,
1
∆(z, µ)
∆(z∗ , ν)
=
= (N((−σx∗ + ρ)2 + (σy∗ )2 )) 2 .
∆(z, λi ) ∆(z∗ , ∞)
If y∗ is replaces by ty∗ where t is a positive scalar factor, then the expression
above is a strictly monotonic increasing function of t, whereas the coordinates
= z∗ = x ∗ +
X1 , . . . , Xn , Y1 , . . . , Yn−1 remain unchanged. Thus, if z ∈ qi and zA−1
i
iy∗ , then for z(t) = (x∗ + ity∗ )Ai , t > 1, the inequalities ∆(z, µ) > ∆(z, λi )(µ , λi )
are satisfied and moreover for a small change in the coordinates X1 , . . . , Xn ,
Y1 , . . . , Yn−1 , the inequalities (150)∗ are also satisfied. As a consequence, if
z ∈ qi , then every neighbourhood of z intersects Ri ; in other words, Ri = qi .
From above, we have that if z ∈ qi , then the entire curve defined by z(t) =
∗
(x + ity∗ )Ai , t ≥ 1 lies in qi and hence for z ∈ Ri , in particular, z(t) for large
t > 0 belongs to Ri , Using this it may be shown that qi and similarly Ri are
connected.
The existence of inner points of q j is an immediate consequence of result
(ii) on p. 199. For j = 1, it may be verified that z = (it, . . . , it), t > 1 is an inner
point of q1 .
Now we proceed to prove that F is a fundamental domain for Γ in Hn . We
have first to show that
(α) the images F M of F for M ∈ Γ cover Hn without gaps.
204
Modular Functions and Algebraic Number Theory
181
Proof. This is obvious from the very method of construction of F. First, for
any z ∈ Hn , there exists a cusp λ such that ∆(z) = ∆(z, λ). Let λ = (λi ) M for
some λi and M ∈ Γ. We have then
∆(z M−1 ) = ∆(z) = ∆(z, λ) = ∆(z M−1 , λi )
and hence z M−1 ∈ Fλi . Now we can find N ∈ Γλi such that (z M−1 )N is reduced
with respect to Γλi and thus zN M−1 ∈ F.
Next, we show that
(β) the images F M of F for M ∈ Γ cover Hn withtout overlaps.
Proof. Let z1 , z2 ∈ F such that z1 = (z2 ) M for an M , ± 10 01 in Γ and let
z1 ∈ qi and z2 ∈ q j . Now, since z1 ∈ Fλi , we have
∆(z1 , λi ) ≤ ∆(z1 , (λ j ) M ) = ∆(z2 , λ j ).
Similarly
∆(z2 , λ j ) ≤ ∆(z2 , (λi ) M−1 ) = ∆(z1 , λi ).
Therefore, we obtain
∆(z1 , λi ) = ∆(z2 , λ j ) = ∆(z1 , (λ j ) M ).
Let us first suppose that ∆(z1 , λi ) = ∆(z2 , λ j ) < d. Then, since ∆(z1 , λi ) < d
as also ∆(z1 , (λ j ) M ) < d, we infer from the result (ii) on p. 199 that λi = (λ j ) M .
But λi and λ j , for i , j are not equivalent with respect to Γ. Therefore i = j
and M ∈ Γλi . Again, since both z1 and z2 are in Gλi and further z1 = (z2 ) M with
M ∈ Γλi , we conclude that necessarily z1 and z2 belong to BdF and indeed their
local coordinates X1 , . . . , Xn , Y1 , . . . , Yn−1 relative to λi satisfy at least one of the
finite number of conditions X1 = ±1/2, . . . , Xn = ±1/2, Y1 = ±1/2, . . . , Yn−1 =
±1/2. Further M clearly belongs to a finite set M1 , . . . , Mr of elements in
h
S
Γλi .
i=1
We have now to deal with the case
d ≤ ∆(z1 , λi ) ≤ c and d ≤ ∆(z2 , λ j ) ≤ c
where we may suppose that λi , (λ j ) M . Let, for i = 1, 2, . . . , h, Bi denote the
set of z ∈ Hn for which d ≤ ∆(z, λi ) ≤ c and z ∈ Gλi . Then, by our remark on p.
205
Modular Functions and Algebraic Number Theory
196 Bi is compact and so is B =
b
h
S
182
Bi . Now, both z2 and z1 = (z2 ) M belong to
i=1
the compact set B. We may then deduce from Proposition 21 that M belongs
to a finite set of elements Mr+1 , . . . , M s in Γ, depending only on B and hence
only on K. Moreover z1 satisfies
206
∆(z1 , λi ) = ∆(z1 , (λ j ) M )
with (λ j ) M , λi . Hence z1 and similarly z2 belongs to BdF. As a result,
arbitrarily near z1 and z2 , there exist points z such that ∆(z, λi ) , ∆(z, (λ j ) M ) for
M = Mr+1 , . . . , M s .
Thus, finally, no two inner points of F can be equivalent with respect to
Γ. Further F intersects only finitely many of its neighbours F M1 , . . . , F Ms and
indeed only on its boundary. We have therefore established (ii).
From (α) and (β) above, it follows that F is a fundamental domain for Γ in
Hn . It consists of h connected ‘pieces’ corresponding to the h inequivalent base
cusps λ1 , . . . , λh and is bounded by a finite number of ‘manifolds’ of the form
∆(z, λi ) = ∆(z, (λ j ) M ), i, j = 1, 2, . . . , h, M = Mr+1 , . . . , M s
(156)
and the hypersurfaces defined by
1
1
Xi(k) = ± , Y (k)
j = ± , i = 1, 2, . . . , n; j = 1, 2, . . . , n − 1
2
2
(k)
where X1(k) , . . . , Xn(k) , Y1(k) , . . . , Yn−1
are local coordinates relative to the base
cusp λk .
The ‘manifolds’ defined by (156) are seen to be generalizations of the ‘isometric circles’, in the sense of Ford, for a fuchsian group. In fact, if λi = ρi /σi
and (λ j ) M = ρ/σ, then the condition (156) is just
N(| − σi z + ρi |) = N(| − σz + ρ|).
If we set n = 1 and λi = ∞ or equivalently ρi = 1 and σi = 0, then the condition
reads as
| − σz + ρ| = 1
which is the familiar ‘isometric circle’ corresponding to the transformation
ηz − ξ
of H1 onto itself.
z→
−σz + ρ
The conditions ∆(z, λ) ≥ ∆(z, λi ) by which Fλi was defined, simply mean
for n = 1 and λi = ∞ that |γz + δ| ≥ 1 for all pairs of coprime rational integers
(γ, δ). Thus, just as the points of the well-known fundamental domain in H1
207
Modular Functions and Algebraic Number Theory
183
for the elliptic modular group lie in the exterior of the the isometric circles
|γz + δ| = 1 corresponding to the same group, Fλi lies in the ‘exterior’ of the
generalized isometric circles ∆(z, λi ) = ∆(z, λ).
We now prove the following important result, which will be used later.
Proposition 22. Let F∗ denote the set of z ∈ F for which ∆(z, λi ) ≥ ei > 0,
i = 1, 2, . . . , h. Then F∗ is compact in Hn .
The proof is almost trivial in the light of our remark on p. 196. Let Bi denote
the set of z ∈ Gλi for which ei ≤ ∆(z, λi ) ≤ c. Then Bi as also B = ∪hi=1 Bi is
compact in Hn . Hence F∗ which is closed and contained in B, is again compact.
Suppose instead of Hn , we consider the space H∗n of (z1 , . . . , zn ) ∈ C n with
Im(z1 ) < 0 and Im(zi ) > 0 for i = 2, . . . , n. Then by means of the mapping
(z1 , . . . , zn ) → (z1 , z2 , . . . , zn ) of hn onto H∗n , F goes over into a fundamental
domain for Γ in H∗n . The general case when we take instead of Hn , the product
of r uppper half-planes and n − r lower half-planes, can be dealt with, in a
similar fashion.
It was Blumenthal who first gave a method of constructing a fundamental
domain for Γ in Hn but his proof contained an error since he obtained a fundamental domain with just one cusp and not h cusps. This error was set right by
Maass.
We remark finally that our method of constructing the fundamental domain
F is essentially different from the well-known method of Fricke for constructing a ‘normal polygon’ for a discontinuous group of analytic automorphisms
of a bounded domain in the complex plane. Our method uses only the notion
of “distance of a point of Hn from a cusp”, whereas we require a metric invariant under the group, for Fricke’s method which is as follows. Let G be a
bounded domain in the complex plane and Γ, a discontinuous group of analytic
automorphisms of G. Further let G possess a metric D(z1 , z2 ) for z1 , z2 ∈ G,
having the property that D((z1 ) M , (z2 ) M ) = D(z1 , z2 ) for M ∈ Γ. We choose a
point z0 which is not a fixed point of any M ∈ Γ, other than the identity and a
fundamental domain for Γ in G is given by the set of points z ∈ G for which
D(z, (z0 ) M ) ≥ D(z, z0 ) for all M ∈ Γ.
208
Now we know that Hn carries a Riemannian metric which is invariant under
Γ. Suppose we use this metric and try to adapt Fricke’s method of constructing
a fundamental domain. For our later purposes, we require a fundamental domain whose nature near the cusps should be well known. Therefore we see in
the first place that the adaptation of Fricke’s method to our case is not practical
in view of the fact that the distance of a point of Hn from the cusps, relative to
Modular Functions and Algebraic Number Theory
184
the Riemannian metric, is infinite. In the second place, it is advantageous to
adapt Fricke’s method only when the fundamental domain is compact, whereas
we know that the fundamental domain is not compact in our case. Moreover,
our method of construction of the fundamental domain uses the deep and intrinsic properties of algebraic number fields.
The following proposition is necessary for our later purposes.
Proposition 23. For any compact set C in Hn , there exists a constant b =
b(C) > 0 such that C ∩ Uµ,b , ∅ for all cusps µ.
Proof. Since C is compact, we can find α, β > 0 depending only on C such that
for z = x + iy ∈ C, we have β ≤ N(y) ≤ α. Now, for any cusp λ = (ρ/σ)(ρ, σ ∈
o) and z = x + iy ∈ C, it is clear that ∆(z, λ) = (N((−σx + ρ)2 y−1 + σ2 y))1/2
satisfies

1
1


|N(σ)|N(y) 2 ≥ β 2 , for σ , 0
∆(z, λ) ≥ 

|N(ρ)|N(y)− 12 ≥ α− 21 , for σ = 0.
1
1
If we choose b for which 0 < b < min(α− 2 , β 2 ), then it is obvious that for all
cusps λ, Uλ,b ∩ C = ∅.
3 Hilbert modular functions
We shall first develop here some properties of (entire) Hilbert modular forms
and, in particular, show that a Hilbert modular form satisfies the regularity condition at the cusps of the fundamental domain F, automatically for n ≥ 2. We
shall then prove the existence of n+1 (entire) modular forms f0 (z), f1 (z), . . . , fn (z)
fn (z)
f1 (z)
,...,
are analytically inof ‘large’ weight k such that the functions
f0 (z)
f0 (z)
dependent. Next, we shall obtain estimates for the dimensions of the space
of entire modular forms of ‘large’ weight. Finally, we shall show that every
Hilbert modular function can be expressed as the quotient of two Hilbert modular forms of the same weight and use this result to prove the main theorem
that the Hilbert modular functions form an algebraic function field of n variables over the field of complex numbers.
Let K be a totally real algebraic number field of degree n over R and let Γ
be the corresponding Hilbert modular group. Further let k be a rational integer.
A complex-valued function f (z) defined in Hn is called an entire Hilbert
modular form of weight k (or of dimension-k) belonging to the group Γ, if
(1) f (z) is regular in Hn ,
(2) for every M = αγ βδ ∈ Γ, f (z M )N(γz + δ)−k = f (z), (??)
209
Modular Functions and Algebraic Number Theory
185
• for every cusp λi = ρi /σi of F, f (z)N(−σi z + ρi )k is regular at the cusp λi
P
2πiS (βzA−1 )
i
where,
i.e. it has a Fourier expansion of the form c(0) + c(β)e
β
in the summation, only terms corresponding to β > 0 can occur.
It will be shown later, as a consequence of a slightly more general result,
that for n ≥ 2, conditions 1) and 2) together imply 3). However, for n = 1, it is
clear that 1) and 2) do not imply 3) and one has necessarily to impose condition
3) also in the definition of an entire modular form.
We shall hereafter refer to entire Hilbert modular forms belonging to Γ
merely as modular forms, without any
of misunderstanding.
risk
−β
Now, if in (??), we take −M = −α
−γ −δ instead of M, we have
f (z M )N(−γz − δ)−k = f (z) = f (z M )N(γz + δ)−k ,
i.e.
f (z M )N(γz + δ)−k ((−1)nk − 1) = 0.
i.e.
210
f (z)((−1)nk − 1) = 0.
Therefore, if nk is odd, necessarily f (z) is identically zero. We are not interested in this case and we shall suppose hereafter that
nk ≡ 0(
mod 2).
Let λ = ρ/σ be a cusp of Hn , a = (ρ, σ) and A =
be the different of K. We then have
ρ ξ
ση
∈ S. Further let ϑ
Theorem 17. Let f (z) be regular in an annular neighbourhood of a cusp λ of
Hn defined by z ∈ Hn , 0 < r < ∆(z, λ) < R and let for every M ∈ Γλ , (??) be
satisfied by f (z). Then, for n ≥ 2, f (z) is regular in 0 < ∆(z, λ) < R and we
have
X
f (z)N(−σz + ρ)k = c(0) +
c(β)e2πiS (βzA−1 ) ,
a2 ϑ−1 |β>0
where the summation is over all β ∈ a2 ϑ−1 with β > 0 and the series converges
absolutely, uniformly over any compact subset of Hn with 0 < ∆(z, λ) < R.
Proof. Let w = zA−1 and g(w) = f (wA ) = f (z). Consider M = αγ βδ = AT A−1 ∈
−1 ζǫ
ǫ being a unit in K and ζ ∈ a−2 . Then z M = (wT )A and
Γλ with T = 0ǫ ζǫ
−1
f (z M ) = g(wT ). Moreover, if h(w) = N(σw + η)−k , then it is easy to verify that
h(wT ) = N(σwT + η)−k
Modular Functions and Algebraic Number Theory
186
= N(ǫ)−k N(γwA + δ)−k N(σw + η)−k
= N(ǫ)−k N(γz + δ)−k h(w).
Let us now define G(w) = g(w)h(w). Then, from f (z M )N(γz + δ)−k = f (z), we
obtain
G(ǫ 2 w + ζ) = G(wT ) = N(ǫ)k G(w)
In particular, we have
G(w + ζ) = G(w)
G(ǫ 2 w) = N(ǫ)k G(w).
(158)
211
Let α1 , . . . , αn be a minimal basis of a−2 . Further, let W = (W1 , . . . , Wn )
where Wi are defined by
wk =
n
X
αi(k) Wi ,
k = 1, 2, . . . , n.
i=1
From (158), we then see that the function H(W) = G(w) has period 1 in each
variable Wi .
Let wk = uk + ivk and Wk = Xk + iYk , for k = 1, 2, . . . , n. Further let D be the
domain in the W-space defined by v1 > 0, . . . , vn > 0, R−2 < v1 , . . . , vn < r−2 .
(For n = 2, v1 and v2 are bounded between the branches of two hyperbolas as
in the figure. The mapping wk → Wk is just an affine transformation of the
w-space). It is clear from the hypothesis that the function H(W) is regular in
D.
Figure, page 211
Under the mapping W → q = (q1 , q2 , . . . , qn ) with q j = e2πiW j ( j = 1, 2, . . . , n)
the domain D goes into a Reinhardt domain D∗ in the q-space and H ∗ (q) =
H(W) is regular in D∗ . If now q∗ ∈ D∗ , we can find a Reinhardt domain V
defined by ak < |qk | < bk , k = 1, 2, . . . , n such that q∗ ∈ V ∈ D∗ . Since H ∗ (q)
P
cr1 ,...,rn qr11 . . . qrnn which
is regular in V, it has a Laurent expansion
−∞<r1 ,...,rn <∞
converges absolutely, uniformly in every compact subset of V. In other words,
for q ∈ V, we have
X
cr1 ,...,rn e2πi(r1 W1 +···+rn Wn ) .
H(W) = H ∗ (q) =
−∞<r1 ,...,rn <∞
∗
Since D is a Reinhardt domain, it is possible to continue H ∗ (q) analytically
from q∗ to any point q ∈ D∗ through a finite chain of overlapping Reinhardt
212
Modular Functions and Algebraic Number Theory
187
domains of the type of V and if, in addition, we use the fact that the Laurent
expansion is unique, we see that for all W ∈ D,
X
H(W) =
cr1 ,...,rn e2πi(r1 W1 +···+rn Wn ) ,
(159)
−∞<r1 ,...,rn <+∞
the series converging absolutely, uniformly over every compact subset of D.
We can find β1 , . . . , βn in K such that S (αi β j ) = δi j (the Kronecker symbol),
for i, j = 1, 2, . . . , n. Then β1 , . . . , βn constitute a minimal basis of a2 ϑ−1 and
when r1 , . . . , rn run over all rational integers from −∞ to +∞ independently,
β = r1 β1 + · · · + rn βn runs over all numbers of the ideal a2 ϑ−1 . We shall denote
n
P
β(i)
cr1 ,...,rn by c(β). Now Wk =
k wi , k = 1, 2, . . . , n, as can be easily verified.
i=1
We then see from (159) that when w lies in the region v1 > 0, . . . , vn > 0,
R−2 < v1 . . . vn < r−2 , we have
X
c(β)e2πiS (βw) .
(160)
G(w) = H(W) =
β∈a2 ϑ−1
If ǫ is a unit in K, then the transformation w → ǫ 2 w corresponds to an
affine transformation of the W-space which takes D into itself. We therefore
have from (160) that
X
2
c(β)e2πiS (βǫ w) .
G(ǫ 2 w) =
β∈a2 ϑ−1
On the other hand, from (158) and (160),
X
G(ǫ 2 w) = N(ǫ)k
c(β)e2πiS (βw) .
β∈a2 ϑ−1
For β ∈ a2 ϑ−1 , βǫ 2 is also in a2 ϑ−1 . On comparing the Fourier coefficients of
G(ǫ 2 w), we obtain
c(βǫ 2 ) = N(ǫ)−k c(β),
i.e.
|c(βǫ 2 )| = |c(β)|.
Thus
|c(βǫ 2m )| = |c(β)| for any rational integer m.
Let us consider a unit cube U in D, defined as the set of W = (W1 , . . . , Wn ),
W j = X j + iY ∗j and 0 ≤ X j ≤ 1. Further let |H(W)| ≤ M on U. From (159), we
have for β = r1 β1 + · · · + rn βn ,
213
Modular Functions and Algebraic Number Theory
c(β) = cr1 ,...,rn =
Z
...
Z
188
H(W)e−2πi(r1 W1 +···+rn Wn ) dX1 . . . dXn
W∈U
∗
= e2πS (βv )
Z
1
...
0
Z
1
H(W)e−2πi(r1 X1 +···+rn Xn ) dX1 . . . dXn ,
0
W∈U
i.e.
∗
|c(β)| ≤ e2πS (βv ) M,
where v∗ = (v∗1 , . . . , v∗n ) with v∗j =
n
P
i=1
α(i j) Yi∗ > 0.
Taking βǫ 2m instead of β, we have
|c(βǫ 2m )| ≤ e2πS (βǫ
i.e.
|c(β)| ≤ e2πS (βǫ
2 −1
2m ∗
v )
2m ∗
v )
M,
M.
(161)
(i)
Let now β ∈ a ϑ with β < 0 for some i. We claim that c(β) = 0,
necessarily. For, using the existence of n − 1(n ≥ 2!) independent units in K
we can find a unit ǫ in K such that
|ǫ (i) | > 1, |ǫ ( j) | < 1 for j , i.
Then, as m tends to infinity, S (βǫ 2m v∗ ) tends to −∞. Hence, from (161), we see
that unless β > 0 or β = 0, c(β) = 0. Thus
X
G(w) = c(0) +
c(β)e2πiS (βw) ,
a2 ϑ−1 |β>0
i.e.
f (z)N(−σz + ρ)k = c(0) +
X
c(β)e2πiS (βzA−1 ) ,
(162)
a2 ϑ−1 |β>0
where the series on the right hand side converges absolutely, uniformly in every
compact subset of the region r < ∆(z, λ) < R.
Let now 0 < ∆(z, λ) ≤ r i.e. r−2 ≤ N(yA−1 ) < ∞. We can then find a
number t with 0 < t < 1 such that R−2 < N(tyA−1 ) < r−2 . Hence the series
P
c(0) +
c(β)e2πiS (βwt) converges absolutely. Now, since
a2 ϑ−1 |β>0
c(0) +
X
a2 ϑ−1 |β>0
c(β)e2πiS (βw) ≤ |c(0)| +
X
β
|c(β)||e2πiS (tβw) |,
214
Modular Functions and Algebraic Number Theory
189
it converges absolutely. It is now clear that the series on the right hand side
of (162) converges absolutely, uniformly over every compact subset in Hn for
which 0 < ∆(z, λ) ≤ r and provides the analytic continuation of f (z)N(−σz +
ρ)+k in 0 < ∆(z, λ) < R. Thus f (z) is regular in the full neighbourhood 0 <
∆(z, λ) < R, of the cusp λ and moreover, in this neighbourhood,
X
f (z)N(−σz + ρ)k = c(0) +
c(β)e2πiS (βzA−1 ) .
(163)
a2 ϑ−1 |β>0
From the theorem above, we can deduce two important corollaries.
(a) Let Vi denote the neighbourhood 0 < ∆(z, λi ) < di < c of the cusp
h
S
λi (i = 1, 2, . . . , h) of F and let F∗ = F − Vi . Then a function f (z)
i=1
which is regular in F∗ and satisfies (??) for all M ∈ Γ is automatically
regular in the whole of F and hence in the whole of hn and is an entire
modular form, for n ≥ 2.
(b) For n ≥ 2, if a function f (z) satisfies conditions (1) and (2) in the definition of a modular form, it satisfies condition (3) automatically and is
therefore regular at the cusp F.
The proofs of (a) and (b) are very simple, in view of Theorem 17.
The second of the two results
√ above was first proved by Gotzky in the
special case n = 2 and K = Q( 5), the class number h being 1. In the general
case, it was proved by Gundlach. An analogue of this result in the case of
modular forms of degree n was established by Koecher.
In the theorem above, we might, instead of Hn , take, for example, the space
H∗n of (z1 , . . . , zn ) ∈ C n with Im(z1 ) < 0, Im(z2 ) > 0, . . . , Im(zn ) > 0. Then,
under the same conditions as in the theorem above, we have for f (z)N(−σz+ρ)k
P
c(β)e2πiS (βzA−1 ) , where β runs over all
an expansion of the form c(0) +
2 −1
(1)
a2 ϑ−1 |β>0
(2)
numbers of a ϑ for which β < 0, β > 0, . . . , β(n) > 0. For other types of
such spaces again we have similar expansions for f (z)N(−σz + ρ)k at the cusp
ρ/σ.
So far, there was no restriction on k except that k should be integral and
kn should be even. In view of the following proposition however, we shall
henceforth be interested only in modular forms of weight k > 0.
Proposition 24. A modular form of weight k < 0 vanishes identically and the
only modular forms of weight 0 are constants.
215
Modular Functions and Algebraic Number Theory
190
Proof. First, let f (z) be a modular form of weight k < 0. Consider ϕ(z) =
N(y)k/2 | f (z)| for z = x + iy ∈ Hn . We shall prove that ϕ(z) is bounded in the
whole of Hn . Let, in fact, λi = ρi /σi be a cusp of F and Vi , the neighbourhood
)k/2 |N(−σi z +
of λi defined by 0 < ∆(z, λi ) < d. Now, since N(y)k/2 = N(yA−1
i
ρi )|k , we have
)k/2 | f (z)||N(−σi z + ρi )|k .
(164)
ϕ(z) = N(yA−1
i
Further, from (163),
f (z)N(σi z + ρi )k = ci (0) +
X
2πiS (βzA−1 )
ci (β)e
i
a2 ϑ−1 |β>0
and clearly for z ∈ Vi ∩ F, f (z)N(−σi z + ρi )k is bounded. Moreover, since
)k/2 < d−k , we conclude from (??) that ϕ(z) is bounded in
for z ∈ Vi , N(yA−1
i
h
h
S
S
(Vi ∩ F). Let F∗ = F − (Vi ∩ F). Since F∗ is compact, ϕ(z) is bounded on
i=1
i=1
F∗ . Thus there exists a constant M such that ϕ(z) ≤ M for z ∈ F. But now for
each P = αγ βδ ∈ Γ, ϕ(zP ) = ϕ(z), since
| f (zP )|N(yP )k/2 = | f (zP )||N(γz + δ)|−k N(y)+k/2 = | f (z)|N(y)k/2
= ϕ(z).
Thus, for all z ∈ hn , ϕ(z) ≤ M and hence
| f (z)| = ϕ(z)N(y)−k/2
≤ MN(y)−k/2 .
Now
f (z) = c(0) +
X
(165)
c(β)e2πiS (βz)
ϑ−1 |β>0
and
Z
1
Z
1
f (z)e−2πiS (αz) dx1 . . . dxn
0
0
Z 1 Z 1
= e2πS (αy)
...
f (x + iy)e−2πiS (αx) dx1 . . . dxn .
c(α) =
...
0
0
Hence, from (165),
|c(α)| ≤ MN(y)−k/2 e2πS (αy)
Letting y tend to (0, . . . , 0) and noting that k < 0, we see that c(α) = 0 for all
α ∈ ϑ−1 and hence f (z) is identically zero.
216
Modular Functions and Algebraic Number Theory
191
Let now ψ(z) be a modular form of weight 0. For each cusp λi of F, we
have
X
2πiS (βzA−1 )
i
.
ψ(z) = ci (0) +
ci (β)e
a2i ϑ−1 |β>0
Consider χ(z) =
h
Q
(ψ(z) − ci (0)). Let, if possible, χ(z) be not identically zero
i=1
i.e. for z0 ∈ F, let |χ(z0 )| = ρ , 0. Since ψ(z) − ci (0) tends to zero as z tends
to λi in F, we can find a neighbourhood Vi of λi such that for z ∈ Vi ∩ F,
h
S
|χ(z)| < (1/2)ρ. Now F∗ = F − (Vi ∩ F) is compact and |χ(z)| attains its
i=1
maximum M at some point of F∗ . We see easily that
M = max |χ(z)| = max |χ(z)|.
z∈F
z∈hn
But his means that χ(z) attains its maximum modulus over Hn at an inner point
of Hn and by the maximum principle, χ(z) is a constant. But now, since χ(z)
tends to zero as z tends to the cusps of F, χ(z) = 0 identically in Hn . This
h
Q
means that (ψ(z) − ci (0)) = 0. Since ψ(z) is single-valued and regular in Hn ,
i=1
it follows that χ(z) is a constant.
Our object now is to give an explicit construction of modular forms of
weight k > 2 (k being integral). Our method will involve an adaptation of the
well-known idea of Poincare for constructing automorphic forms belonging to
a discontinuous group Γ0 of analytic automorphisms of the unit disc D defined
by |z| < 1 in the complex z-plane. Now, any analytic automorphism of D
az + b
, where a and b are complex numbers
is given by the mapping z →
bz + a
satisfying |a|2 − |b|2 = 1. Let the elements of Γ0 be Mi = abii baii , i = 1, 2, . . .
ai z + bi
of D).
bi z + ai
Let z0 be a given point of D. Now, since Γ0 acts discontinuously on D, we
can find a neighbourhood V of z0 such that the images V M of V for M ∈ Γ0 do
not intersect V, unless (z0 ) M = z0 . Let n0 be the order of the isotropy group of
z0 in Γ0 . Then we see that
X"
dx dy ≤ n0 · π.
(corresponding to the automorphisms z → z Mi =
M∈Γ0 V
M
217
Modular Functions and Algebraic Number Theory
192
On the other hand, for M = ba ba ∈ Γ0 , we have
"
"
dx dy =
|bz + a|−4 dx dy.
V
VM
X"
Therefore,
M∈Γ0 V
218
−4
|bz + a| dx dy ≤ n0 · π.
(166)
Since |a|2 − |b|2 = 1, a , 0 and further |b/a| < 1. Hence, for z ∈ D,
bz + a < 1 + |z|.
1 − |z| ≤ a (167)
Let V be contained in the disc |z| ≤ r < 1 and let v denote the area of V. We
have, then, in view of (166) and (167),
X"
X
|a|−4 ≤
v(1 + r)−4
|bz + a|−4 dx dy ≤ n0 · π.
(168)
M∈Γ0 V
M∈Γ0
P
We conclude that
M∈Γ0
z ∈ V,
Hence
P
X
M∈Γ0
|a|−4 is convergent. Further, from (167), we have for
|bz + a|−4 < (1 − r)−4
X
M∈Γ0
|a|−4
(bz + a)−4 converges absolutely, uniformly on every compact subset
M∈Γ0
of V. Since z0 is arbitrary, this is true even for every compact subset of D.
P
(bz + a)−k
Using the same idea as above, we shall prove that actually
M∈Γ0
converges absolutely for k > 2. Consider
"
dx dy
;
Il =
(1 − |z|2 )l
V
the integral converges if and only if l < 1. Now, as before
X"
dx dy
≤ n0 Il .
(1
− |z|2 )−l
M
VM
But
"
VM
dx dy
=
(1 − |z|2 )l
"
V
|bz + a|−4+2l
!
dx dy
a b
for
M
=
∈ Γ0 .
b a
(1 − |z|2 )l
Modular Functions and Algebraic Number Theory
Thus
X"
M∈Γ0 V
|bz + a|2l−4
dx dy
=
(1 − |z|2 )l
X"
M V
M
193
219
dx dy
≤ n0 Il .
(1 − |z|2 )l
P
(bz + a)−4+2l
Using the same argument as earlier, it is easily shown that
M∈Γ0
P
(bz + a)−k , (for k > 2) converges absolutely, uniformly on
and therefore
M∈Γ0
every compact subset of D.
Let k be an even rational integer greater than 2. Then we see that ψ(z, k) =
P
(bz + a)−k is analytic in D. To verify that ψ(z, k) is an automorphic form
M∈Γ0
belonging to Γ0 , we have only to show that
!
p q
for N =
∈ Γ0 , ψ(zN , k)(qz + p)−k = ψ(z, k).
q p
Now, for
!
!k/2
dz M
a b
∈ Γ0 ,
M=
= (bz + a)−k
b a
dz
and so
ψ(z, k) =
X dz M !k/2
dz
M∈Γ
0
Further
X dz MN !k/2
ψ(zN , k) =
dzN
M∈Γ0
!−k/2 X
!k/2
dz MN
dzN
=
dz
dz
M∈Γ
0
k
= (qz + p) ψ(z, k),
since MN runs over all elements of Γ0 , when M does so. Thus ψ(z, k) is an
automorphic form of weight k, for the group Γ0 and is a so-called Poincaré
series of weight k for Γ0 .
The method given above for the construction of automorphic forms for the
group Γ0 is due to Poincaré and it cannot be carried over as such to the case
of the Hilbert modular group, since the euclidean volume of Hn is infinite. In
the following, we shall give a method of constructing Hilbert modular forms of
220
Modular Functions and Algebraic Number Theory
194
weight k > 2, based on a slight modification of the idea of Poincaré. We shall
be interested only in the case n ≥ 2.
Let λ = ρ/σ be a cusp of Hn , a = (ρ, σ) = ai (for some i), A = σρ ηξ ∈ S
associated with λ and zA−1 = z∗ = x∗ + iy∗ . We had introduced in Hn a sys1
tem of ‘local coordinates’ relative to the cusp λ, namely, N(y∗ )− 2 , Y1 , . . . , Yn−1 ,
X1 , . . . , Xn . Defining q = N(y∗ ), it is easily seen that the relationship between
the volume element dv = dq, dY1 , . . . , dYn−1 , dX1 , dX2 , . . . , dXn and the ordinary euclidean volume element dx1 dy1 , . . . , dxn dyn in Hn is given by
dv =
N(a)2
√ |N(−σz + ρ)|−4 dx1 dy1 , . . . , dxn dyn ,
2n−1 R0 |d|
where R0 and d are respectively the regulator and the discriminant of K. Let
now
N(a)2
c=
√
n−1
2 R0 |d|
and let dω be the volume element N(y)−2 dx1 dy1 , . . . , dxn dyn . Then dω is invariant under a (modular) substitution z → z M for M ∈ Γ and moreover
dv = c · N(y∗ )2 dω = c∆(z, λ)−4 dω.
Now, under the transformation z → z M for M ∈ Γ, it may be verified that
∆(z, λ)4
,
∆(z, λ M−1 )4
→ ∆(z M , λ)−2 = ∆(z, λ M−1 )−2 .
dv → c∆(z M , λ)−4 dω =
N(y∗ ) = ∆(z, λ)−2
(169)
We can find a point z0 Uaλ,d such that z0 is not a fixed point of any modular
transformation except the identity. Now, by Theorem 16 again, we can find a
neighbourhood
V of z0 contained
in Uλ,d ∩ Gλ such that V ∩ V M = ∅, except
for M = 10 01 in Γ. Let M1 = 10 01 , M2 , . . ., be a complete system of representatives of the right cosets of Γ modulo Γλ . For i , 1, Mi takes Uλ,d into Uµ,d
where µ = λ Mi , λ and since Uλ,d ∩ Uµ,d = ∅, we see that for i , 1, V Mi lies
wholly outside the neighbourhood Uλ,d . Further V Mi ∩V M j = ∅ unless Mi = M j .
Now, by applying suitable transformations in Γλ to the images V Mi or parts of
the same, we might suppose that V Mi ⊂ Gλ for all i. It is to be remarked that
under this process, the sets V Mi continue to have finite euclidean volume and to
be mutually non-intersecting. Let now, for a suitable d0 > 0, V ⊂ Uλ,d − Uλ,d0 .
We then see that the mutually disjoint sets V Mi , i = 1, 2, . . . , are all contained
in Gλ − Uλ,d0 . It is now easily verified that
Z
XZ
. . . N(y∗ ) s−2 dv
Js =
Mi
V Mi
221
Modular Functions and Algebraic Number Theory
≤
=
Z
Z
d0
0
d0
Z
1
2
...
− 21
Z
1
2
− 21
195
q s−2 dq dY1 , . . . , dYn−1 dX1 , . . . , dXn
q s−2 dq
0
and J s converges if and only if s > 1.
On the other hand, we obtain by (169), for M = αγ βδ ∈ Γ
Z
Z
Z
Z
|N(−σz + ρ)|4 N(y) s−2
∗ s−2
dv.
. . . N(y ) dv =
...
|N(−σz M + ρ)|2s |N(γz + δ)|2s
VM
V
Now, let
∗
A Mj = Pj =
γj
−1
!
∗
,
δj
j = 1, 2, . . .
Then P j runs over a complete set of matrices of the form A−1 M with M ∈ Γ
such that for i , j, Pi , NP j with N ∈ A−1 Γλ A. Equivalently, we see that
(γ j , δ j ) runs over a complete set of non-associated pairs of integers γ j , δ j in a
such that (γ j , δ j ) = a. Now
Z
XZ
. . . N(y∗ ) s−2 dv
Js =
Mi
=
X′ Z
γ j ,δ j
VM
...
V
Z
|N(γ j z + δ j )|−2s |N(−σz + ρ)|2s N(y∗ ) s−2 dv
where on the right hand side, the summation is over a complete set of nonassociated pairs of integers (γ j , δ j ) such that (γ j , δ j ) = a.
Let B be a compact set in Hn . We can then find constants c13 and c14
depending only on B such that for all z ∈ B and real µ and ν, we have
c13 |N(µ j + ν)| ≤ |N(µz + ν)| ≤ c14 |N(µ j + ν)|.
(170)
Taking for B, the closure V and V in Hn , we see that for a suitable constant c15
depending only on V and for s > 1,
X′
|N(γ j i + δ j )|−2s ≤ c15 J s < ∞.
γ j ,δ j
Let now z lie in an arbitrary compact set B in Hn . Then, in view of (170),
we can find a constant c16 depending only on B, such that for all z ∈ B,
X′
X′
|N(γ j z + δ j )|−2s ≤ c16
|N(γ j i + δ j )|−2s < ∞.
γ j ,δ j
γ j ,δ j
222
Modular Functions and Algebraic Number Theory
Thus the series
P′
196
N(γ j z + δ j )−2s converges absolutely, uniformly over
γ j ,δ j
every compact subset of Hn , for s > 1.
Let a1 and a2 denote a full system of pairs of integers (γ, δ) satisfying
(γ, δ) = a and not mutually associated, respectively with respect to the group
of all units in K and the group of units of norm 1 in K.
Let k be a rational integer greater than 2. Corresponding to an integral ideal
a in K, we now define for z ∈ Hn , the function
X
N(γ∗ z + δ∗ )−k .
G∗ (z, k, a) =
(γ∗ ,δ∗ )∈a2
We first observe that G∗ (z, k, a) is well-defined and is independent of the particular choice of elements of a2 . Further, since the group of units of norm 1 in
K is of index at most 2 in the group of all units in K. We deduce from the foreP
going considerations that the series
N(γ∗ z + δ∗ )−k converges absolutely
(γ∗ ,δ∗ )∈a2
uniformly over any compact subset of Hn . Hence G∗ (z, k, a) is regular in Hn .
Moreover, for M = µρ νx ∈ Γ,
X
N(γ∗ z M + δ∗ )−k
G∗ (z M , k, a) =
(γ∗ ,δ∗ )∈a2
= N(µz + ν)k
X
N((γ∗ ρ + δ∗ µ)z + γ∗ x + δ∗ ν)−k .
(γ∗ ,δ∗ )∈a2
Now (γ∗ ρ + δ∗ µ, γ∗ x + δ∗ ν) again runs over a system a2 when (γ∗ , δ∗ ) does so.
Thus
G∗ (z M , k, a)N(µz + ν)−k = G∗ (z, k, a).
Therefore G∗ (z, k, a) is a modular form of weight k.
Let k be odd and K contain a unit ǫ0 of norm −1; for example, if the degree
n is odd, then −1 has norm −1. If (γ, δ) runs over a system a1 , then (γ, δ)
together with (ǫ0 γ, ǫ0 δ) runs over a system a2 . Thus
X
N(γz + δ)−k = 0,
G∗ (z, k, a) = (1 + N(ǫ0 )k )
(γ,δ)∈a1
since N(ǫ0 )k + 1 = 0. Thus the functon G∗ (z, k, a) vanishes identically in the
case when k is odd and K contains a unit of norm −1. We shall henceforth
exclude this case from our consideration; in particular, nk should necessarily
be even. We have then
G∗ (z, k, a) = ρG(z, k, a),
223
Modular Functions and Algebraic Number Theory
197
where, by definition
G(z, k, a) =
X
N(γz + δ)−k
(γ,δ)∈a1
and ρ is the index of the group of units of norm 1 in K in the group of all units
in K. The functions G(z, k, a) are again modular forms of weight k and are the
so-called Eisenstein series for the Hilbert modular group.
If a = (µ)ai for some i(= 1, 2, . . . h) and µ ∈ K, then G(z, k, a) = N(µ)−k G(z, k, a224
i ).
For the sake of brevity, we may denote G(z, k, ai ) by Gi (z, k) for i = 1, 2, . . . , h.
We can obtain the Fourier expansion of Gi (z, k) at the various cusps λ j
of F (cf. formula (163) above) by using the following generalized ‘Lipschitz
formula’ valid for rational integral s > 2 and τ ∈ Hn , namely,
!ns
2π
X
X
i
N(τ + ζ)−s = √
(N(µ)) s−1 e2πS (µτ) .
(171)
n
N(ϑ)(Γ(s))
−1
ζ integral
µ∈ϑ
ζ∈K
µ>0
But we shall not go into the details here.
A modular form f (z) is said to vanish at a cusp λ = ρ/σ of Hn , if, in the
Fourier expansion of f (z)N(−σz + ρ)k at the cups λ (cf. formula (163)), the
constant term c(0) is zero. If f (z) vanishes at a cusp λ, it clearly vanishes at all
the equivalent cusps.
We shall now prove the following
Proposition 25. The Eisenstein series Gi (z, k)(k > 2) vanishes at all cusps of
Hn except at those equivalent to λi .
Proof. Let λ j = ρ j /σ j be a base cusp of F and A j = σρ jj ηξ jj be the associated
matrix in S. Then setting w = zA−1j , we have
Gi (z, k)N(−σ j z + ρ j )k =
X′
N((γρ j + δσ j )w + γξ j + δη j )−k .
(γ,δ)=ai
Let us denote by ci j the constant term in the Fourier expansion of Gi (z, k)N(−σ j z+
√
√
ρ j )k at the cusp λ j . If we set w = ( −1t, . . . −1t)(t > 0), then
ci j = lim Gi (z, k)N(−σ j z + ρ j )k
t→∞
X′
√
= lim
N((γρ j + δσ j ) −1t + γξ j + δη j )−k
t→∞
(γ,δ)=ai
Modular Functions and Algebraic Number Theory
=
X′
(γ,δ)=ai
198
√
lim N((γρ j + δσ j ) −1t + γξ j + δη j )−k ,
t→∞
(172)
since the interchange of the limit and the summation can be easily justified.
Now unless γρ j + δσ j = 0, the corresponding term in the series (172) is zero.
If γρ j + δσ j = 0, then
(γ) (ρ j ) (δ) (σ j )
=
ai a j
ai a j
225
and from this, it follows that necessarily i = j. Further, if γρ j + δσ j = 0, it can
be easily shown that γξ j + δη j , 0. Moreover, in the series (172), there can
occur only the pair (γ, δ) for which γρ j + δσ j = 0 and indeed this happens only
if − = j. Therefore ci j = 0, unless i = j. Our contention that Gi (z, k) vanishes
at all cusps of F except at λi is therefore established.
We deduce that the h Eisenstein series Gi (z, k), i = 1, 2, . . . , h are linearly
h
P
independent over the field of complex numbers. If αiGi (z, k) = 0, then letting
i=1
!
h
P
z tend to cusp λ j , the relation
αiGi (z, k) N(−σ j z + ρ j )k = 0 gives α j c j j = 0
i=1
i.e. α j = 0. Thus αi = 0 for i = 1, 2, . . . , h.
A modular form which vanishes at all the cusps of F is known as a cusp
form.
It is now easy to prove
Proposition 26. For a given modular form f (z) of integral weight k > 2, we
can find a linear combination ϕ(z) of the Eisenstein series Gi (z, k) such that
f (z) − ϕ(z) is a cusp form.
Proof. If c j (0) is the constant term in the Fourier expansion of f (z)N(−σ j z +
h
P
ci (0)c−1
ρ j )k at cusp λ j = ρ j /σ j of F, then the function ϕ(z) =
ii G i (z, k)
i=1
satisfies the requirements of the proposition.
The method outlined above for constructing modular forms yields only
Eisenstein series but no cusp forms. In order to construct nontrivial cusp forms
of integral weight k > 2, we proceed as follows.
Let H∗n denote the set of z ∈ C n for which −z ∈ Hn . Further let w = u − iv ∈
∗
Hn . We define for z = x + iy ∈ Hn , integral l and integral k > 2, the function
X
Φ(w, z) =
N(γz + δ)−k N(w − z M )−l
(173)
M∈Γ
226
Modular Functions and Algebraic Number Theory
199
where M = αγ βδ runs over all elements of Γ. We shall show that the series
(173) converges absolutely, uniformly for z and w lying in compact sets in Hn
and H∗n respectively, provided that l > 2, k ≥ 4.
To this end, we choose
z0 ∈ R1 such that z0 is not a fixed point of any M
in Γ, except M = 10 01 . We can then find a neighbourhood V of z0 such that
the images V M of V for M , 10 01 in Γ, do not intersect V. We may suppose
moreover that for a suitable d′ > 0, if W denotes the annular neighbourhood
of the base cusp ∞ defined by 0 < d′ < ∆(z, ∞) < d, then V ⊂ G∞ ∩ W. It
will then follow from result (ii) on p. 199 that for M < Γ∞ , V M lies outside the
neighbourhood U∞,d and moreover for M ∈ Γ∞ , V M ⊂ W. In other words, for
S
VM ,
M ∈ Γ, the images V M of V are mutually disjoint and further if z ∈
M∈Γ
S
V M is contained in a region of
then necessarily we have ∆(z, ∞) ≥ d′ , i.e.
M∈Γ
the form N(y) < α, for some α > 0.
Let us now consider the integral
Z
Z
I(s, l, α) =
. . . N(y) s−2 |N(w − z)|−l dv,
N(y)≤α
where dv = dx1 dy1 , . . . , dxn dyn and the integral is extended over the set in Hn
defined by N(y) ≤ α. If s ≥ 2, then trivially
Z
Z
s−2
. . . |N(w − z)|−l dv
I(s, l, α) ≤ α
Hn
i.e.
I(s, l, α) ≤ α s−2
= α s−2
Z
< ∞,
Z
Z
...
Hn
∞
Z
...
0
Z
∞
...
−∞
N((u − x)2 + (v + y)2 )−l/2 dv
Z
∞
N(v + y)1−l dy1 . . . dyn
0
∞
N(1 + x2 )−l/2 dx1 . . . dxn
−∞
provided l > 2. Now, since the sets V M are mutually disjoint, we have
!
Z
XZ
k
. . . N(y)k/2−2 |N(w − z)|−l dv ≤ I , l, α < ∞,
2
M∈Γ
VM
227
Modular Functions and Algebraic Number Theory
if l > 2, k ≥ 4. But, on the other hand, for M =
Z
Z
. . . N(y)k/2−2 (N(w − z))−l dv
VM
Z
...
XZ
...
=
V
α β
γ δ
200
∈ Γ,
Z
N(y)k/2−2 (N(w − z M ))−l N(γz + δ)−k dv
Z
N(y)k/2−2 |N(w − z M )|−l |N(γz + δ)|−k dv
Hence
M∈Γ
V
!
k
≤ I , l, α < ∞.
2
(174)
Let B, B∗ be compact sets in Hn and H∗n respectively. Then, for z ∈ B and
w ∈ B∗ we can find constants c17 and c18 depending only on B and B∗ and on
z0 , such that
0 < c17 ≤
P
M∈Γ
|N(w − z M )|−l |N(γz + δ)|−k
≤ c18 .
|N(w − (z0 ) M )−l N(γz0 + δ)|−k
(175)
Taking for B, the closure V of V in Hn , we obtain from (174) and (175) that
|N(w − (z0 ) M )|−l |N(γz0 + δ)|−k converges and indeed uniformly with respect
to w ∈ B∗ . But again from (175), we have for z ∈ B and w ∈ B∗ ,
X
|N(w − z M )|−l |N(γz + δ)|−k
M∈Γ
≤ c18
X
M∈Γ
|N(w − (z0 ) M )|−l |N(γz0 + δ)|−k .
Hence for l > 2, k ≥ 4, the series (173) converges absolutely, uniformly when
z and w lie in compact sets in Hn and H∗n respectively.
We shall suppose hereafter that l > 2, k ≥ 4; then we see that Φ(w, z) is
∗
regular in H
nα∗asβ∗ a function of z and regular in Hn , as a function of w.
If P = γ∗ δ∗ ∈ Γ, then we have
X
Φ(w, zP ) =
N(γzP + δ)−k N(w − z MP )−l
M∈Γ
N(γ∗ z + δ∗ )k
X
M∈Γ
N((γα∗ + δγ∗ )z + γβ∗ + δδ∗ )−k N(w − z MP )−l
228
Modular Functions and Algebraic Number Theory
201
i.e.
−1
Φ(w, zP )N(γ∗ z + δ∗ )−k = Φ(w, z).
(176)
ǫ ζǫ
0 ǫ −1
∈ Γ with a unit ǫ and integral ζ in K, then
X
Φ(wT , ζ) =
N(γz + δ)−k N(ǫ 2 w + ζ − z M )−l
Further, if T =
M∈Γ
= N(ǫ)k
X
M∈Γ
N(ǫγz + ǫδ)−k N(w − zT − 1 M )−l
= N(ǫ)k Φ(w, z).
In particular, Φ(w + ζ, z) = Φ(w, z) for integral ζ ∈ K. We shall now proceed to
obtain the Fourier expansion of Φ(w, z).
Let Γt denote the subgroup of Γ consisting of M = 10 1ζ with integral ζ in
K. Then clearly
X
X
N(w − z M + ζ)−l ,
(177)
Φ(w, z) =
N(γz + δ)−k
ζ integral
ζ∈K
M∈Γt \Γ
where, in the outer summation, M runs over a full set of representatives of the
right cosets of Γ modulo Γt . It is now easy to see that the set {±M|M ∈ Γt \Γ}
is in one-one correspondence with the set of all pairs of coprime integers (γ, δ)
in K. To each coprime pair (γ, δ) of integers in K, let us assign a fixed M =
∗ ∗
γ δ ∈ Γ. Then
X
X
N(w − z M + ζ)−l .
2Φ(w, z) =
N(γz + δ)−k
(γ,δ)=(1)
ζ integral
ζ∈K
The inner sum is obviously independent of the particular M ∈ Γ associated
with the pair (γ, δ).
In view of (??), we have
X
N(w − z M + ζ)−l
ζ integral
ζ∈K
(2πi)ln
= √
|d|(Γ(l))n
X
(178)
l −2πiS (λ(w−z M ))
N(λ) e
.
λ∈ϑ−1 ,λ>0
Therefore, after suitable rearrangement of the series which is certainly permissible, we obtain from (177) and (178)
X
(2πi)ln
Φ(w, z) = √
N(λ)l e−2πiS (λw) ×
2 |d|(Γ(l))n λ∈ϑ−1 ,λ>0
229
Modular Functions and Algebraic Number Theory
×
X
202
e2πiS (λzM ) N(γz + δ)−k .
(λ,δ)=0
We now define for k > 2 and z ∈ Hn ,
X
fλ (z, k) =
e2πiS (λzM ) N(γz + δ)−k .
(179)
(γ,δ)=0
Then we have
Φ(w, z) =
(2πi)ln
√
2 |d|(Γ(l))n
X
N(λ)l e−2πiS (λw) fλ (z, k).
λ∈ϑ−1 ,λ>0
Now, the series (179) converges absolutely, uniformly over every compact
subset of Hn , since the series (173) does so. Therefore fλ (z, k) is regular in
Hn ; further
∗ ∗ it is bounded in the fundamental domain, for n = 1. Moreover, for
P = αγ∗ βδ∗ ∈ Γ, we have from (176),
fλ (zP , k)N(γ∗ z + δ∗ )−k − fλ (z, k),
in view of the uniqueness of the Fourier expansion of Φ(w, z). Thus fλ (z, k) is
a modular form of weight k and in fact, a cusp form, since it might be shown
that as z tends to the cusp λi = ρi /σi in F, fλ (z, k) · N(−σi z + ρi )k tends to
zero. A priori, it might happen that fλ (z, k) = 0 identically for all λ ∈ ϑ−1
with λ > 0. However, we shall presently see that this does not happen for large
k. The series (179) are called Poincaré series for the Hilbert modular group.
They were introduced
by Poincaré in the case n = 1 (elliptic modular group).
Let Ai = σρii ηξii ∈ S correspond to the cusp λi = ρi /σi of F and let ai =
(ρi , σi ). By considering the function
X
Φ(w, z, Ai ) =
N(γz + δ)−k N(w − zP )−l
P
∗ ∗
where P = γ δ runs over all elements of S of the form A−1
i M with M ∈ Γ, we
can obtain the Poincaré series
X
N(γz + δ)−k e2πiS (λzP ) ,
(180)
fλ (z, k, Ai ) =
(γ,δ)=ai
which are again cusp forms of weight k. In (180), the summation is over all
pairs of integers (γ, δ) such that (γ, δ) = ai and with each such pari (γ, δ) is
associated a fixed P = γ∗ ∗δ ∈ S, of the form A−1
i M with M ∈ Γ. Further λ > 0
2 −1
is a number in ai ϑ . It is clear that the Poincaré series fλ (z, k, A1 ) coincide
230
Modular Functions and Algebraic Number Theory
203
with fλ (z, k) introduced above. Poincaré theta-series for a discontinuous group
of automorphisms of a bounded domain in C n have been discussed by Hua.
The Fourier coefficients of the Poincaré series fλ (z, k, Ai ) have been explicitly determined by Gundlach and they are expressible as infinite series involving Kloosterman sums and Bessel functions.
It has been shown by Maass that for fixed i(= 1, 2, . . . , h) the Poincaré
series fλ (z, k, Ai ) for λ ∈ a2i ϑ−1 , λ > 0 together with the h Eisenstein series
Gi (z, k) generate the vector space of modular forms of weight k(> 2) over the
field of complex numbers. The Poincaré series fλ (z, k, Ai ) themselves generate
the subspace of all cusp forms of weight k; this is the so-called ‘completeness
theorem for the Poincaré series’. The essential idea of the proof is the introduction of a scalar product ( f, ϕ) for two modular forms f (z) and ϕ(z) of
weight k, of which at least one is a cusp form, namely,
Z
Z
( f, ϕ) =
...
f (z)ϕ(z)N(y)k−2 dx1 dy1 , . . . , dxn dyn
F
the integral being extended over a fundamental domain for the Hilbert modular
group. This scalar product which is known as the “Petersson scalar product”
was used by Petersson for n = 1, in order to prove the “completeness theorem”
for the Poincaré series of real weight k ≥ 2, belonging to even more general
groups than the elliptic modular group, such as the Grenzkreis groups of the
first type. They are called horocyclic groups. The “completeness theorem”
for the Poincaré series belonging to the principal congruence subgroups of the
elliptic modular group was proved, independently of Petersson, by A. Selberg
by a different method.
For n = 1, the scalar product ( f, ϕ) has been explicitly computed by Rankin.
As we remarked earlier, it could happen a priori that all the Poincaré series
fλ (z, k) vanish identically. However, we shall presently see that this does not
happen if k is large.
Proposition 27. There exist n + 1 modular forms f0 (z), f1 (z), . . . , fn (z) of large
weight k such that f1 (z) f0−1 (z), . . . , fn (z) f0−1 (z) are analytically independent.
Proof. Let us choose a basis ω1 (= 1), ω2 , . . . , ωn of K over Q with integral
ωi > 0(i = 2, . . . , n); clearly |(ω(i j) )| , 0. Further let λi = ωi + g for a rational
integer g > 0, to be chosen later; again we have |(λ(i j) )| , 0. Let us, moreover,
denote G1 (z, k), fλ1 (z, k), . . . , fλn (z, k) by f0 (z), f1 (z), . . . , fn (z) respectively. We
shall show that for large k and g to be chosen suitably, the n functions gi (z) =
fi (z) f0−1 (z), i = 1, 2, . . . , n, are analytically independent.
231
Modular Functions and Algebraic Number Theory
204
We know that for real t > 1, z0 = (it, . . . , it) is an inner point of q1 . Hence
for all pairs of coprime integers (γ, δ) in K with γ , 0, we have |N(γz0 +δ)| > 1.
P
lim N(γz0 +
One can show easily as a consequence that lim G1 (z0 , k) =
k→∞
232
(γ,δ)=1
δ)−k = 1. Thus, for large k, | f0 (z0 ) − 1| can be made arbitrarily small. Similarly,
P
2
by taking k to be suitably large, we can ensure that | fm (z0 ) − e−2πtS (λm ǫ ) |
ǫ
P
2
and hence |gm (z0 ) − e−2πtS (λm ǫ ) | for m = 1, 2, . . . , n can be made as small as
ǫ
we like; here, in the summation ǫ runs over all units of K. Since gm (z) and
P 2πiS (λm ǫ 2 z)
e
are regular in a neighborhood of z0 we see that
ǫ
∂g (z) X ∂ e2πiS (λm ǫ 2 z) m −
∂z j
∂z
j
ǫ
, m, j = 1, 2, . . . , n,
z=z0
can be made arbitrarily small, by choosing k large.
Now, if ǫ1 , . . . , ǫn are any n totally positive units of K, then the equation
n
n
P
P
S ((ωi + g)ǫi ) = S ((ωi + g)) does not hold for large g > 0, unless we have
i=1
ǫ1 =
i=1
ǫ2 = . . . = ǫn = 1. This result can be easily proved by using the fact that
if ǫ > 0 is a unit of K, S (ǫ) > n unless ǫ = 1. (This last-mentioned fact is
itself a consquence of the well-known result that the arithmetic mean of n(≥ 2)
positive numbers not all equal, strictly exceeds their geometric mean). Thus, if
we consider the jacobian


X ∂ e2πiS (λm ǫ 2 z) 


J(λ1 , . . . , λm , t) = 

∂z
j
z=z
ǫ
0
as a function of t(> 1), then it contains the term (2πi)n 2n |(λ(mj) )|e−2πt
n
P
S (λ M )
m=1
which is not cancelled by any other term. Now, since |(λ(mj) )| , 0, it follows that
for some t > 1, J(λ1 , . . . , λm , t) , 0 and hence, by the foregoing, the jacobian
|(∂gm /∂z j )| does not vanish identically if k is chosen to be large enough. Thus
our proposition is proved.
Incidentally, in the course of the proof above, we have shown that for large
k, the Poincaré series fλ (z, k), λ ∈ ϑ−1 , λ > 0 cannot all vanish identically.
We might sketch here two alternative proofs of Proposition 27. The following proof is shorter and does not use the Fourier expansion of Φ(w, z) introduced earlier.
Now, by (176) and in view of the fact that (for n = 1) lim Φ(w, it) = 0,
t→∞
the function Φ(w, z) is a modular form of weight k (in z). Let now w(p) =
233
Modular Functions and Algebraic Number Theory
∗
(w1(p) , . . . , w(p)
n ) ∈ Hn , P = 1, 2, . . . , n and ΦP (z) =
205
Φ(w(p) , z)
. Then, if z =
G1 (z, k)
(it, . . . , it), t > 1, we see as before that
X
lim ΦP (z) =
N(w(p) − ǫ 2 z + ζ)−l ,
k→∞
ǫ,ζ
where, in the summation ǫ runs over all units of K and ζ over all integers in K.
If we suppose that w(p)
q and zq are all purely imaginary, then for integral ζ , 0
in K and for any unit ǫ, we have
(p)
(q)2 (q)2
(q) w(p)
−
ǫ
z
<
w
−
ǫ
z
−
ζ
q
, p, q = 1, 2, . . . , n.
q
q
q
If we further suppose that iw(p)
q and zq /2i, p, q = 1, 2, . . . , n lie close to 1
and use some well-known inequalities for elementary symmetric functions of
n positive real numbers, we can show that for a unit ǫ , ±1,
|N(w(p) − z)| < |N(w(p) − ǫ 2 z)|,
p = 1, 2, . . . , n.
It now follows that

! 


2
∂Φ
(z)

P
= 

lim lim l−n N(w(1) − z)l . . . N(w(n) − z)l ∂zq  w(p) − zq 
l→∞ k→∞
q
and for suitable choice of w(1) , . . . , w(n) , we can ensure that




1
 (p)
 , 0.
wq − zq Thus, for large k and l, the n functions Φ1 (z), . . . , Φn (z) are analytically independent.
One can also prove Proposition 27 as follows. First, by the map z =
zj − θj
(z1 , . . . , zn ) → w = (w1 , . . . , wn ) where w j =
and θ = (θ1 , . . . , θn ) ∈ F.
zj − θj
One maps Hn analytically homeomorphically D, the unit disc in C n defined by
|w1 | < 1, . . . , |wn | < 1. Now the group Γ induces a discontinuous group of
analytic automorphisms of D. By using the well-known methods for a discontinuous group of analytic automorphisms of a bounded domain in C n , we can
find n + 1 Poincaré series f0 (w), f1 (w), . . . , fn (w) of weight k and belonging
f1 (w)
fn (w)
to G such that the n functions
,...,
are analytically independent.
f0 (w)
f0 (w)
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Modular Functions and Algebraic Number Theory
206
Pulling these functions back to Hn by the inverse map, we obtain the desired
n + 1. Hilbert modular forms of weight k.
The modular forms of weight k form a vector space over the field of complex numbers. We shall see that this vector space is finite-dimensional and its
dimension satisfies certain estimates in terms of k, for large k. In the following,
we shall be interested only when k > 2.
Theorem 18. Let t(k) denote the number of linearly independent modular
forms of weight k. Then for large k(> 2),
0 < c19 ≤
t(k)
≤ c20 ,
kn
(181)
for constants c19 and c20 depending only on K.
(Note. Since for k > 2, t(k) ≥ 1, (181) may be upheld for all k > 2, by
choosing c19 and c20 properly).
Proof. We shall first prove the relatively easier inequality t(k) ≥ c19 kn .
By the preceding proposition, there exist n+1 modular forms f0 (z), f1 (z), . . . , fn (z)
of weight j such that f1 (z) f0−1 (z), . . . , fn (z) f0−1 (z) are analytically independent
and therefore algebraically independent. Let us now suppose that k > n j + 2
and let m be the smallest positive integer such that
mn j + 2 ≤ k ≤ (m + 1)n j + 2.
(182)
Consider the power-products f1k1 (z) f2k2 (z), . . . , fnkn (z), (ki = 0, 1, 2, . . . , m).
These are (m + 1)n in number. And the (m + 1)n functions ( f0 (z))mn−(k1 +···+kn ) ,
f1k1 (z), . . . , fnkn (z)(ki = 0, 1, 2, . . . , m) are modular forms of weight mn j and are
linearly independent, since f1 (z) f0−1 (z), . . . , fn (z) f0−1 (z) are algebraically independent. Since k > mn j + 2, we can obtain, by multiplying these modular
forms by an Eisenstein series G1 (z, r) (where r may be chosen to be k − mn j),
(m + 1)n modular forms of weight k. Thus
t(k) ≥ (m + 1)n .
By (182), (m + 1)n ≥ ((k − 2)/n j)n . Thus, for large k, we can find a constant
c19 > 0 such that t(k) ≥ c19 kn .
We now proceed to prove the other inequality in (181).
Let a be an ideal in K and p, a positive rational integer. Then it is known
that for a constant c21 depending only on a and K, the number of β ∈ a for
235
Modular Functions and Algebraic Number Theory
207
which β > 0 or β = 0 and S (β) < p is at most equal to c21 pn . Taking a to
be one of the h ideals ϑ−1 , a22 ϑ−1 , . . . , a2h ϑ−1 , we can find a constant c22 > 0
depending only on K such that the number of β ∈ a2i ϑ−1 (i = 1, 2, . . . , h) for
which S (β) < p and β > 0 or β = 0 is at most c22 pn . Let c23 = hc22 .
Suppose f1 (z), . . . , fq (z) are q linearly independent modular forms of weight
k(> 2). We shall be interested only in the case when k is so large as to allow
the situation q > c23 . Let us therefore suppose that q > c23 . Let p be the largest
positive rational integer such that
c23 pn < q ≤ c23 (p + 1)n .
Now each one of the modular forms f j (z), j = 1, 2, . . . , q has at the cusp
λi = ρi /σi , a Fourier expansion of the form
X
2πiS (βzA−1 )
i
.
f j (z)N(−σi z + ρi )k = c(i)
(0)
+
c(i)
j
j (β)e
β∈a2i ϑ−1 ,β>0
We can find complex numbers µ1 , . . . , µq not all zero such that for f (z) =
µ1 f1 (z) + · · · + µq fq (z), all the Fourier coefficients c( j) (β) for all β ∈ a2j ϑ−1
such that β = 0 or β > 0 and S (β) < p, in the expansion
X
2πiS (βzA−1 )
j ,
(183)
f (z)N(−σ j z + ρ j )k = c( j) (0) +
c( j) (β)e
a2j ϑ−1 |β>0
vanish. This is possible, since it only involves solving r(≤ c23 pn ) linear equations in q(> c23 pn ) unknowns µ1 , . . . , µq . Clearly, f (z) is a cusp form of weight
k.
Let z = x + iy ∈ Fλ j ∩ Gλ j and yA−1j = (y∗1 , . . . , y∗n ). Then we know that
c7 ≤ q
n
y∗i
N(yA−1j )
≤ c8
(see p. 253) and therefore as z ∈ F tends to the cusp λ j , each one of y∗1 , . . . , y∗n
tends to infinity. Since c( j) (0) = 0, it follows that
N(y)k/2 | f (z)| = N(yA−1j )k/2 |N(−σ j z + ρ j )|k | f (z)| → 0,
as z ∈ F tends to any cusp λ j of F.
k/2
Let us now define for z ∈ Hn , the function g(z) = N(y)
| f (z)|. This function is a non-analytic modular function i.e. for M = αγ βδ ∈ Γ,
g(z M ) = N(y M )k/2 | f (z M )| = N(y)k/2 |N(γz + δ)|−k | f (z M )| = g(z).
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Modular Functions and Algebraic Number Theory
208
By the foregoing, g(z) tends to zero as z tends to the cusps of F and as a consequence, if we now make use of Proposition 22, we see that g(z) is bounded
in F and attains its maximum µ > 0 at a point z0 = x0 + iy0 ∈ F. But, since
g(z M ) = g(z) for all M ∈ Γ, we obtain
g(z) ≤ g(z0 ) = µ,
(184)
for all z ∈ Hn . By multiplying f (z) if necessary by a number of absolute value
1, we might suppose that f (z0 )N(y0 )k/2 = µ.
Let us assume without loss of generality that z0 ∈ Fλi ∩ Gλ1 (otherwise, if
0
z ∈ Fλ j ∩ Gλ j , j , 1, then we have only to work with the coordinates zA−1j and
z0A−1 instead of z and z0 ). Let t1 = u1 + iv1 be a complex variable and let us
j
denote the n-tuple (t1 , t1 , . . . , t1 ) by t = u + iv. Then for
v1 < s =
inf y0 ,
k=1,2,...,n k
we see that z = z0 − t ∈ Hn .
Setting r = e−2πit1 , we now define for |r| < e2πs , the function h(r) = f (z0 −
2πipt1
t)e
= f (z)e2πipt1 . Taking j = 1 in (183), we obtain
X
0
h(r) = r−p
c(1) (β)e2πiS (βz ) rS (β) .
(185)
ϑ−1 |β>0
S (β)≥p
The right hand side of (185) is a power series in r containing only non-negative
powers of r and is absolutely convergent for |r| < R = e2πs > 1. Thus h(r) is
regular for |r| < R. Let now 1 < R1 = e2πv1 < R. Then, by the maximum
modulus principle,
h(1) ≤ max |h(r)|,
|r|=R1
i.e.
f (z0 ) ≤ max | f (z0 − t)|e−2πpv1
|r|=R1
i.e.
µN(y0 )−k/2 ≤ µN(y0 − v)−k/2 e−2πpv1 ,
in view of (184). From this, we have however
k
log N(1/(1 − v1 )/y0 )
2
k
= v1 S (1/y0 ) + · · · terms involving higher powers of v1
2
2πpv1 ≤
237
Modular Functions and Algebraic Number Theory
209
i.e.
k
S (1/y0 ) + · · · terms involving v1 and higher powers.
4π
Now letting R1 tend to 1 or equivalently v1 to zero, we obtain
p≤
p≤
k
S (1/y0 ).
4π
From p. 196, we see that S > c9 (for a constant c9 depending only on K)
and S (1/y0 ) ≤ n · c9 . (We have similar inequalities for z0A−1 , if z0 ∈ Fλ j ∩ Gλ j ,
j
j , 1). Thus, in any case, we can find a constant c24 depending only on K such
that
p ≤ c24 k.
Therefore, we see that if there exist q linearly independent modular forms
of weight k, then necessarily
q ≤ c23 (p + 1)n ≤ c23 (c24 k + 1)n ≤ c20 kn
for large k and a suitable constant c20 independent of k. In particular, we have
t(k) ≤ c20 kn
which proves the theorem completely.
The idea used in the proof above is essentially the same as in Siegel’s proof
of the analogous result that the maximal number of linearly independent modular forms of degree n and weight k is at most equal to c25 k(n/2)(n+1) for a suitable
constant c25 independent of k. (See reference (28) Siegel, p. 642).
We now proceed to consider the Hilbert modular functions.
A function f (z) meromorphic in Hn is called a Hilbert modular function
(or briefly, a modular function), if
(i) for every M ∈ Γ, f (z M ) = f (z);
(ii) for n = 1, f (z) has at most a pole at the infinite cusp of the fundamental
domain.
For n ≥ 2, no condition is imposed on the behaviour of f (z) at the cusps of
F, in the definition of a modular function.
If n = 1, f (z) is just an elliptic modular function. Let us remark, however,
that for n ≥ 2, there is no analogue of the elliptic modular invariant j(τ). For, if
f (z) is a Hilbert modular function regular in Hn (n ≥ 2) then it is automatically
regular at the cusps of F in view of Theorem 17 and is, in fact, a modular form
of weight 0 and hence a constant.
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Modular Functions and Algebraic Number Theory
210
The modular functions constitute a field which contains the field of complex numbers. Our object now is to study the structure of this field. We shall
not be concerned, in the sequel, with the case n = 1, since, in this case, it is
well-known that the elliptic modular functions form a field of rational functions
generated by j(τ) over the field of complex numbers.
In the case n ≥ 2, we shall see presently that the modular functions form an
algebraic function field of n variables. The proof which we are going to present
is mainly due to Gundlach.
A major step in the direction of this result is the following
239
Lemma . For n ≥ 2, every Hilbert modular function can be expressed as the
quotient of two Hilbert modular forms of the same weight k.
Proof. Let F∗ = F −
h
S
i=1
Uλi ,d with 0 < d′ < d. Then we know that F∗ is
compact, in view of Proposition 22.
Let ϕ(z) be a Hilbert modular function. Suppose we can find a modular
∗
form χ(z) of weight
α β k such that ψ(z) = ϕ(z)χ(z) is regular in F . Then, since
−k
for every M = γ δ ∈ Γ, ψ(z M )N(γz + δ) = ψ(z) and further since ψ(z) is
regular in Uλi ,d − Uλi ,d (for i = 1, 2, . . . , h), we can appeal to Corollary (a) (p.
214) and conclude that ψ(z) is a modular form of weight k. Thus ϕ(z) would be
expressible as the quotient of the two modular forms ψ(z) and χ(z) of weight
k.
Our object therefore is to construct a modular form χ(z) of weight k such
that χ(z) annihilates the poles of ϕ(z) in F∗ i.e. more precisely, such that χ(z)ϕ(z)
is regular in F∗ .
To every point z0 ∈ F∗ , we assign a neighbourhood R in Hn such that the
following conditions are satisfied
P(z1 − z01 , . . . , zn − z0n )
P(z − z0 )
in R where P and Q are
=
Q(z − z0 )
Q(z1 − z01 , . . . , zn − z0n )
convergent power-series in z − z0 and locally coprime at every point of
R.
(i) ϕ(z) =
(ii) By a suitable linear transformation, we can find new variables w, v2 , . . . , vn
such that by the Weierstrass’ preparation theorem, we can suppose that in
R, Q(z1 −z01 , . . . , zn −z0n )−wg +ai (v)wg−1 +· · ·+ag (v) where a1 (v), . . . , ag (v)
are power-series in v2 , . . . , vn convergent in R. If ϕ(z) is regular in R, then
we might take Q to be 1 in R.
(iii) Further, we might assume that R is a polycylinder of the form |w| ≤ b,
|v2 | ≤ a, . . . , |vn | ≤ a where a and b are so chosen that, in addition, any
240
Modular Functions and Algebraic Number Theory
211
zero (w, v2 , . . . , vn ) of Q(w, v2 , . . . , vn ) for which |v2 | < a, . . . , |vn | < a,
automatically lies in R. We shall denote the (n − 1)-tuple (v2 , . . . , vn ) by
v.
Let R∗ ⊂ R be the corresponding neighbourhood of z0 ∈ F∗ defined by
|w| < b, |v2 | < a/2, . . . , |vn | < a/2. Then the neighbourhoods R∗ for z0 ∈ F∗
give an open covering of the compact set F∗ and we extract therefrom a finite
′
covering of F∗ by neighbourhoods R∗1 , . . . , R∗c′ of points z(1) , . . . , z(c ) of F∗ .
′
Let R1 , . . . , Rc′ be the corresponding neighbourhoods of z(1) , . . . , z(c ) defined
above.
gi
P
(i)
v
a(i)
Let in Ri , ϕ(z) = P(w, v)/Q(w, v) where Q(w, v) =
v (v)w and av (v)
v=0
are convergent power series in v2 , . . . , vn in Ri . If ϕ(z) is regular in Ri , then
c′
P
Q(w, v) = 1 in Ri and therefore gi = 0. Let us denote gi by c23 . The constant
i=1
c26 of course depends on ϕ(z).
If M is the set of poles of ϕ(z), then M∩R is just the set of zeros of Q(z−z0 )
in R. Clearly, M is invariant under the modular substitutions. Let us denote
M ∩ F∗ by N; since F∗ is compact, so is N.
Let k > 2 be a rational integer to be chosen suitably later. From Theorem
18, we know that the number r = t(k) − h of linearly independent cusp forms
of weight k satisfies r > c27 kn for a constant c27 > 0, if k > c∗27 (a suitable
constant). Let f1 (z), f2 (z), . . . , fr (z) be a complete set of linearly independent
cusp forms of weight k.
Now, let f (z) = g(w, v2 , . . . , vn ) be regular in Ri ; then, performing the division of g(w, v2 , . . . , vn ) by Q(w, v2 , . . . , vn ), we can write in Ri ,
f (z) = h(w, v2 , . . . , vn )Q(w, v2 , . . . , vn ) +
gX
i −1
s
b(i)
s (v2 , . . . , vn )w ,
(187)
s=0
where h(w, v2 , . . . , vn ) is regular in Ri and b(i)
s (v2 , . . . , vn ) are convergent powerseries in v2 , . . . , vn .
Let l be the non-negative rational integer such that
c26 ln−1 < c27 kn ≤ c26 (l + 1)n−1 .
(188)
We might assume k to be so large that l ≥ 1. We now claim we can find complex
numbers α1 , . . . , αr not all zero, such that for f (z) = α1 f1 (z) + · · · + αr fr (z),
the power-series b(i)
s (v2 , . . . , vn ), occurring in (187) are of order at least l. in
v2 , . . . , vn , for i = 1, 2, . . . , c′ . This is simple to establish, for it only involves
solving q(≤ c26 ln−1 ) linear equations in r(> c27 kn ) unknowns α1 , . . . , αr and by
(188), q < r.
241
Modular Functions and Algebraic Number Theory
r
P
212
We shall first prove that for all sufficiently large k, the cusp form f (z) =
αi fi (z) chosen above, vanishes identically on N. Let, if possible, f (z) be
i=1
not identically zero on N. Then f (z) attains its maximum modulus µ , 0 at
some point z∗ on N, since N is compact. We shall suppose that f (z∗ ) = 1,
after normalizing f (z) suitably, if necessary. Now z∗ ∈ R∗i for some i. Let w∗ ,
v∗2 , . . . , v∗n be the coordinates of z∗ , in terms of the new variables w, v2 , . . . , vn
in Ri . Let t be a complex variable. Denoting the (n − 1)-tuple (v∗2 t, . . . , v∗n t) by
v∗ t, we have (w, v∗ t) ∈ Ri , for |w| < b, |t| ≤ 2.
Figure, page 241
Let Mi = M ∩ Ri . Consider now the analytic set defined in the polycylinder
|w| < b, |t| ≤ 2 by Q(w, v∗ t) = 0. Let B be an irreducible component of this
analytic set, containing (w∗ , 1). Clearly if (w, t) ∈ B, then (w, v∗ t) ∈ Mi . On B,
therefore, we have after (187), for the function f0 (w, v∗ t) = f (z) that
f ∗ (w, t) = f0 (w, v∗ t) = tl
gX
i −1
∗
v
t−1 b(i)
v (v t)w .
v=0
−l (i) ∗
Since b(i)
v (v2 , . . . , vn ) is of order at least l in v2 , . . . , vn , it follows that t bv (v t)
−l ∗
is regular in |t| ≤ 2. Hence t f (w, t) is regular on B. Now it can be shown
that if (w0 , t0 ) ∈ B, then B can be parametrized locally at (w0 , t0 ) by
t − t0 = uq , w − w0 = R(u),
where q is a positive integer and R(u) is regular in u. If we note that we can
apply the maximum modulus principle to the function t−l f ∗ (w, t) as a function
of u, we can show that t−l f ∗ (w, t) attains its maximum modulus on B at a point
(w(0) , t(0) ) on the boundary of B and hence with |t(0) | = 2, necessarily. Thus
1 = f (z∗ ) = f0 (w∗ , v∗ ) ≤ Max |t−l f ∗ (w, t)|
B
−l
= 2 | f ∗ (w(0) , t(0) )|
≤ 2−l Max | f ∗ (w, v∗ t)|
|t|≤2,|w|≤b
(w,t)∈B
(189)
≤ 2−l Max | f (z)|.
Mi
Let z(0) be a point on the boundary of Mi at which max | f (z)| is attained. If
z∈Mi
z(0) ∈ F∗ , then z(0) ∈ N and hence | f (z(0) )| ≤ 1. Hence we have 1 ≤ 2−l , which
gives a contradiction, since l ≥ 1. Therefore f (z) must vanish on N.
242
Modular Functions and Algebraic Number Theory
213
Suppose now z(0) < F∗ . Then, for some M ∈ Γ, we can ensure that z(0)
M ∈ F.
We contend that there are only finitely many P ∈ Γ for which FP intersects
c′
c′
S
S
Ri . To prove this, we first apply Proposition 23 with Rl with C and then
i=1
i=1
for a certain b = b(C) > 0, we have Uµ,b ∩ C = ∅ for all cusps µ. It follows that
for all cusps λ and for all P ∈ Γ, we have (Uλ,b )P ∩ C = ∅. In particular, for
h
S
λ = λ1 , . . . , λh , we see that (Uλ,b )P ∩ C = ∅, for all P ∈ Γ. Now F − Uλi ,b is
i=1
compact and we apply Proposition 21 with F −
h
S
for B and
λi ,b
c′
S
Ri for B′ . Then
i=1
we see immediately that our assertion above is true.
We may now suppose that z(0)
M ∈ F ∩ Uλ1 ,c , without loss of generality. Since
c′
S
z(0) ∈ Ri , and since by the remark above, M = αγ βδ belongs to a finite set in
i=1
Γ independent of z(0) , there exists a constant c28 > 1 such that |N(γz(0) + δ)| >
(0)
(0)
+ δ)|−k = | f (z(0) )|. Hence from (189), we have
c−1
28 . Further | f (z M )||N(γz
1 ≤ 2−l | f (z(0) )| ≤ 2−l ck28 | f (z(0)
M )|.
(0)
∗
If z(0)
M ∈ F , then we have | f (z M )| ≤ 1 and therefore
1 ≤ 2−l ck28 .
(0)
∗
′
Suppose now z(0)
M < F ; then z M ∈ Uλ1 ,d ∩ F. We shall show again that
≤ 1.
Let M∗ be the connected component of M ∩ Uλ1 ,d′ which contains the point
(0)
(i)
z M . Then f (z) is not locally constant on M∗ near z(0)
M , because of f (z M ) = 0
and the connectedness of Ri .
S
M∗T then clearly sup | f (z)| = δ, again. Now,
Let sup | f (z)| = δ. If B =
| f (z(0)
M )|
z∈M∗
T ∈Γλl
z∈B
since f (z) tends to zero as z tends to the cusp λ1 , there exists d′′ with 0 < d′′ <
d′ such that | f (z)| < (1/2)δ in Uλ1 ,d′′ . Let B1 = B ∩ (Uλ1 , d′ − Ulλ1 , d′′ ) ∩ F;
S
then B ∩ (Uλ1 ,d′ − Uλ1 ,d′′ ) =
(B1 )T and further sup | f (z)| = δ, again. Since
T ∈Γλ1
z∈B1
B1 is compact, | f (z)| attains its maximum δ (< ∞, necessarily therefore) at a
point z′ , say, in Bi . Now, there exists a sequence of points w(i) ∈ M∗Ti ∩ F
converging to z′ . Using the fact that M is closed and locally connected, it can
be shown that z′ belongs already to M∗T for a T ∈ Γλ1 , which is one of the
(0)
(0)
T i ’s above. Now | f ((z(0)
M )T )| = | f (z M )| and to show that | f (z M )| ≤ 1, we might
∗
∗
as well argue with MT instead of M . Thus we may assume, without loss of
243
Modular Functions and Algebraic Number Theory
214
generality, that | f (z)| attains its maximum δ on M∗ at a point z′ . We now claim
that ∆(z′ , λ1 ) = d′ . For, if ∆(z′ , λ1 ) < d′ , then z′ ∈ M∗ and we can find a
′
′′
curve C in M∗ connecting z(0)
M with z . Let z be the first point on C such that
′′
| f (z )| = δ; then f (z) is not locally constant on M∗ near z′′ , and the maximum
principle gives a contradiction. Thus ∆(z′ , λ1 ) = d′ . Again, for a suitable
′
′
T ∈ Γλ1 , z′T ∈ F∗ i.e. z′T ∈ N, as before. Thus | f (z(0)
M )| ≤ | f (z )| = | f (zR )| ≤ 1.
Thus, in all cases
1 ≤ ck28 2−l ,
244
i.e.
l ≤ c29 k,
if f (z) does not vanish identically on N. But, from this and from (188), we
obtain
0 < c30 ln < c27 kn ≤ c26 (l + 1)n−1 ≤ c31 ln−1 ,
i.e.
0 < c30 < c27
k
l
!n
≤
c31
≤ c31 .
l
From the fact that (k/l)n is bounded, we see that as k tends to infinity, l also
tends to infinity. But this contradicts the inequality l < c31 /c30 . Thus f (z)
vanishes on N identically, for large k.
Now taking each Ri , i = 1, 2, . . . , c′ , we observe that since f (z) vanishes on
the analytic set defined by Q(z − z0 ) = 0 in Ri , there exists an integer ni such
that f ni (z) is divisible by Q(z − z0 ) in Ri i.e. f ni (z)ϕ(z) is regular in Ri . (We
might take ni = gi , for example). Choosing m = c26 , we see that f m (z)ϕ(z)
is regular on F∗ . Thus the modular form f m (z) of weight mk has the required
property and our lemma is therefore completely proved.
We are now, in a position, to prove (for n ≥ 2), the main theorem, viz.
Theorem 19. The Hilbert modular functions form an algebraic function field
of n variables.
Proof. We know by Proposition 27 that there exist n + 1 modular forms f0 (z),
fn (z)
f1 (z)
,...,
are algebraically indef1 (z), . . . , fn (z) of weight j such that
f0 (z)
f0 (z)
pendent over the field of complex numbers. Since the quotient of two modular
forms of the same weight is a modular function, we have n algebraically independent modular functions
g1 (z) =
fn (z)
f1 (z)
, . . . , gn (z) =
.
f0 (z)
f0 (z)
245
Modular Functions and Algebraic Number Theory
215
Let Ω denote the field of Hilbert modular functions and Ω0 , the field generated by g1 (z), . . . , gn (z) over the field of complex numbers. We shall first show
that every element of Ω satisfies a polynomial equation of bounded degree,
with coefficients in Ω0 .
Let ϕ(z) be any modular function. By the lemma above, ϕ(z) = ψ(z)/χ(z),
where χ(z) and ψ(z) are two modular forms of the same weight k. Let us consider for fixed positive rational integers m and s, the power products



ki = 0, 1, 2, . . . , m,
k1
kn
l
fk1 ,...,kn ,l (z) = g1 (z), . . . , gn (z)ϕ (z), 

l = 0, 1, 2, . . . , s.
Now f0mn (z)χ s (z) fk1 ,...,kn ,l (z) is a modular form of weight w = mn j + ks. We have
indeed (m + 1)n (s + 1) such modular forms of weight w. If we can choose m
and s such that
(m + 1)n (s + 1) > c20 wn (≤ t(w)),
(190)
then we get a nontrivial linear relation between these modular forms and hence
between the (m + 1)n (s + 1) functions fk1 ,...,kn ,l (z). But this linear relation cannot
be totally devoid of terms involving powers of ϕ(z), since otherwise, this will
contradict the algebraic independence of g1 (z), . . . , gn (z). Thus ϕ(z) satisfies a
polynomial equation of degree at most s and with coefficients in Ω0 . Since Ω
is a separable algebraic extension of Ω0 and since every element of Ω satisfies
a polynomial equation of degree at most s over Ω0 , it follows that Ω is a finite
algebraic extension of Ω0 i.e. an algebraic function field of n variables.
In order to complete the proof of the theorem, we have only to find rational
integers m and s which satisfy (190), s being bounded. This is very simple, for
wn
when m tends to infinity,
tends to ( jn)n so that if we choose m large
(m + 1)n
enough and s > c20 jn nn , then (190) will be fulfilled. Our theorem is therefore
completely proved.
By adopting methods similar to the above, Siegel has recently proved that
(for n ≥ 2) every modular function of degree n (i.e. a complex-valued function
meromorphic in the Siegel half-plane and invariant under the modular transformations) can be expressed as the quotient of two “entire” modular forms of
degree n. (See reference (32), Siegel). It had been already shown by Siegel
that the modular functions of degree n which are quotients of “entire” modular
forms, form an algebraic function field of n(n + 1)/2 variables.
246
Bibliography
[1] A. A. Albert: A solution of the principal problem in the theory of Riemann matrices, Ann. of Math., 35 (1934), 500-515.
[2] O. Blumenthal: Über Modulfunktionen von mehreren Veränderlichen,
Math. Ann., 56 (1903), 509-548; ibid 58 (1904), 497-527.
[3] L. R. Ford: Automorphic functions, Chelsea, New York (1951).
[4] F. Götzky: Über eine zahlentheoretische Anwendung von Modulfunktionen zweier Veränderlicher, Math. Ann. 100 (1928), 411-437.
[5] K. B. Gundlach: Über die Darstellung der ganzen Spitzenformen zu den
Idealstufen der Hilbertschen Modulgruppe und die Abschätzung ihrer
Fourierkoeffizienten, Acta. Math. 92 (1954), 309-345.
[6] ——: Poincaresche und Eisensteinsche Reihen Zur Hilbertschen Modulgruppe, Math. Zeit. 64 (1956), 339-352.
[7] ——: Modulfunktionen zur Hilbertschen Modulgruppe und ihre Darstellung als Quotienten ganzer Modulformen, Arch. Math. 7 (1956), 333-338.
[8] ——: Über den Rang der Schar der ganzen automorphen Formen zu hyperabelschen Transformationsgruppen in zwei Variablen, Gött. Nachr. Nr.
3. (1958).
[9] ——: Quotientenraum und meromorphe Funktionen zur Hilbertschen
Modulgruppe, Gött. Nachr. Nr. 3, (1960).
[10] E. Hecke: Grundlagen einer Theorie der Integralgruppen und der Integralperioden bei den Normalteilern der Modulgruppe, Math. Werke,
Göttingen, (1959), 731-772.
216
247
BIBLIOGRAPHY
217
[11] L. Hua: On the theory of Fuchsian functions of several variables, Ann. of
Math., 47 (1946), 167-191.
[12] H. D. Kloosterman: Theorie der Eisensteinschen Reihen von mehreren
Veränderlichen, Abh. Math. Sem Hamb. Univ. 6 (1928), 163-188.
[13] M. Moecher: Zur Theorie der Modulformen n-ten Grades I, Math. Zeit.
59 (1954), 399-416.
[14] S. Lefschetz: On certain numerical invariants of algebraic varieties with
applications to Abelian varieties, Trans. Amer. Math. Soc., 22 (1921),
327-482.
[15] H. Maass: Über Gruppen von hyperabelschen Transformationen, SitzBer. der Heidelb. Akad. der Wiss., math.-nat. K1. 2 Abh. (1940).
[16] —–: Zur Theorie der automorphen Funktionen von n Veränderlichen,
Math. Ann., 117 (1940), 538-578.
[17] ——: Theorie der Poincaréschen Riehen zu den hyperbolischen Fixpunktsystemem der Hilbertschen Modulgruppe, Math. Ann. 118 (1941), 518543.
[18] H. Petersson: Einheitliche Begründung der Vollständigkeitssätze für die
Poincaréschen Reihen von reeler Dimension bei beliebigen Grenzkreisgruppen von erster Art, Abh. Math. Sem. Hamb. Univ. 14 (1941), 22-60.
[19] ——: Über die Berechnung der Skalarprodukte ganzer Modulformen,
Comment. Math. Helv. 22 (1949), 168-199.
[20] E. Picard: Sur les fonctions hyperabéliennes, J. de Liouville, Ser. IV, 1
(1885).
[21] H. Poincaré: Sur les fonctions fuchsiennes, Oeuvres, II, 169-257.
[22] ——: Fonctions modularies et fonctions fuchsiennes, Oeuvres, II, 592618.
[23] R. A. Rankin: The scalar product of modular forms, Proc. Lond. Math.
Soc. 3, II (1952), 198-217.
[24] C. Rosati: Sulle matrici di Riemann, Rendiconti del Circ. Mathematico di
Palermo, 53 (1929), 79-134.
[25] A. Selberg: Beweis eines Darstellungssatzes aus der Theorie der ganzen
Modulformen, Arch. for Math. Naturvid. 44, Nr. 3 (1940).
248
BIBLIOGRAPHY
218
[26] I. I. Piatetskii-Shapiro: Singular modular functions, Izv. Akad. Nauk
SSSR, Ser. Mat. 20 (1956), 53-98.
[27] C. L. Siegel: Über die analytische Theorie der quadratischen Formen, III,
Ann. of Math. 38 (1937), 213-292.
[28] ——: Einführung in die Theorie der Modulfunktionen n-ten Grades,
Math. Ann. 116 (1939), 617-657.
[29] ——: Die Modulgruppe in einer einfachen involutorischen Algebra,
Festschrift Akad. Wiss. Göttingen, 1951.
[30] ——: Automorphe Funktionen in mehreren Variablen, Göttingen, (Function theory in 3 Volumes) (1954-55).
[31] ——: Zur Vorgeschichte des Eulerschen Additons-theorem, Sammelband
Leonhard Euler, Akad. Verlag, Berlin, (1959), 315-317.
[32] ——: Über die algebraische Abhängigkeit von Modulfunktionen n-ten
Grades, Gött. Nacher. Nr. 12, (1960).
[33] H. Weyl: On generalized Riemann matrices, Ann. of Math. 35 (1934),
714-729.
Appendix
Evaluation of Zeta Functions for
Integral Values of Arguments
By Carl Ludwig Siegel
Eulerpresumably enjoyed the summation of the series of reciprocals of
squares of natural numbers in the same way as Leibnitz did the determination of the sum of the series which was later named after him. In the nineteenth
century, generalizations of these results to the L-series introduced by Dirichlet
were studied; these were important for the class number formulae of cyclotomic fields; corresponding questions for the L-series of imaginary quadratic
fields got elucidated by Kronecker’s limit formulae.
In one of his most beautiful papers, Hecke has obtained an analogous result
for the L-series of real quadratic fields, which led him to the assertion that the
value of the zeta function of every ideal class of a real quadratic number field
of arbitrary discriminant d √is, for the values s = 2, 4, 6, . . . , of the variable,
equal to the product of π2s d and a rational number. He further gave a brief
indication of two possible methods of proof for his assertion. The first of these
has recently been carried out in different ways [1]. The second method of proof
leads, after suitable modifications to the generalization, published by Klingen
[2], to arbitrary totally real algebraic number fields. This second method is
indeed quite clear in its underlying general ideas but, however, in the form
available till now, it admits of no effective determination of the said rational
number. By means of a theorem on elliptic modular forms, which in itself
seems not uninteresting, we shall, in the sequel, present a proof by a constructive procedure meeting the usual requirement of Number Theory.
219
249
BIBLIOGRAPHY
220
1 Elliptic Modular Forms
Let z be a complex variable in the upper half plane and
Fk (z) =
1 X′
(lz + m)−k (k = 4, 6, 8, . . .),
2 l,m
(1)
where l, m run over all pairs of rational integers except 0, 0. If we set
X
lg = σg (n) (g = 0, 1, 2, . . .), e2πiz = q
250
t|n
then for this Eisenstein series we have the Fourier expansion
Fk (z) = ζ(k) +
(2πi)k X
σk−1 (n)qn
(k − 1)! n=1
∞
(2)
which, in view of
ζ(k) = −
can also be expressed by the formulae
Fk (z) = ζ(k)Gk (z), Gk (z) = 1 −
(2πi)k Bk
,
2 · k!
(3)
∞
2k X
σk−1 (n)qn (k = 4, 6, . . .).
Bk n=1
(4)
In particular, here, we have the Bernoulli numbers
B4 = −
1
1
5
7
1
, B6 =
, B8 = − , B10 =
, B14 = ;
30
42
30
66
6
and hence
G4 = 1 + 240
∞
X
σ3 (n)qn , G6 = 1 − 504
G8 = 1 + 480
∞
X
σ7 (n)qn , G10 = 1 − 264
n=1
G14 = 1 − 24
n=1
∞
X
∞
X
σ5 (n)qn
n=1
∞
X
σ9 (n)qn
n=1
σ13 (n)qn
n=1
with rational integral coefficients and constant term 1. We set further
G0 = 1.
(5)
BIBLIOGRAPHY
221
Let h be a non-negative even rational integer. If we denote the dimensionof the
linear space Mh of elliptic modular forms of weight h by rh , where for fixed h
we write, for brevity, also r for rh , then it is well known that
" #
" #
h
h
+ 1(h . ( mod 12)), rh =
(h ≡ 2( mod 12)).
rh =
12
12
251
We have, the particular,
G24 = G8 , G4G6 = G10 , G24G6 = G14
GlG14−l = G14 (l = h − 12rh + 12 = 0, 4, 6, 8, 10, 14)
(6)
and for the modular form
∆=q
∞
Y
(1 − qn )24
n=1
of weight 12,
1728∆ = G34 − G26 .
If
j(z) = G34 ∆−1 = q−1 + · · ·
(7)
denotes the absolute invariant, then
∆2
!
dj
dG4
d∆
1
dG4
dG6
,
= 3G24
∆ − G34
=
G24G6 2G4
− 3G6
dz
dz
dz
1728
dz
dz
and the expression in the brackets yields a modular form of weight 12 and
indeed a cusp form which can therefore differ from ∆ at most by a constant
factor. Comparing the coefficients of q in the Fourier expansions, we get
dj
= −G14 ∆−1 .
d log q
(8)
Let hereafter, h > 2 and hence rh > 0. The r isobaric power-products Ga4Gb6
where the exponents a, b run over all non-negative rational integer solutions of
4a + 6b = h,
form a basis of Mh . It follows from this that, for every function M in Mh ,
r−1
MG−1
is a modular form of weight
h−12r+12 always belongs to M12r−12 . Since ∆
12r − 12, not vanishing anywhere in the interior of the upper half-plane,
1−r
=W
MG−1
h−12r+12 ∆
(9)
252
BIBLIOGRAPHY
222
is an entire modular function and hence a polynomial in j with constant coefficients.
Let
T h = G12r−h+2 ∆−r
(10)
with the Fourier expansion
T h = Chr q−r + · · · + Ch1 q−1 + Ch0 + · · ·
(11)
and first coefficient Chr = 1. Since
∆−1 = q−1
∞
Y
n=1
(1 + qn + q2n + · · · )24 ,
(12)
all the Fourier coefficients of T h turn out to be rational integers.
Theorem 1. Let
M = a0 + a1 q + a2 q2 + · · ·
be the Fourier series of a modular form M of weight h. Then
Ch0 a0 + Ch1 a1 + · · · + Chr ar = 0
Proof. For l = 0, 1, 2, . . .
1 d jl+1
dj
=
dz l + 1 dz
and hence, by (??), it has a Fourier series without constant term. Since the
dj
function W defined by (9) is a polynomial in j, the product W
is, similarly,
dz
a Fourier series without constant term.
jl
Becauseof (6) and (8), we have
−
253
dj
1
1−r
W
= MG−1
G14 ∆−1 = MG12r−h+2 ∆−r = MT h
h−12r+12 ∆
2πi dz
from which the theorem follows on substituting the series (11) and (??) for T h
and M respectively.
Let us put for brevity,
ch0 = ch .
It is important, for the entire sequel, to show that ch does not vanish.
Theorem 2. We have
ch , 0.
BIBLIOGRAPHY
223
Proof. First, let h ≡ 2( mod 4), so that h ≡ 2t( mod 12) with t = 1, 3, 5.
Then correspondingly 12r = h − 2, h + 6, h + 2; hence 12r − h + 2 = 0, 8, 4 and
G12r−h+2 = G0 , G24 , G4 .
Since by (5), G4 has all its Fourier coefficients positive and the same holds for
∆−5 as a consequence of (12), we conclude from (10) that all the coefficients in
the expansion (11) are positive. Therefore in the present case, the integers ch0 ,
ch1 , . . . , chr are all positive and, in particular, ch = ch0 > 0 i.e. ch , 0.
Let now h ≡ 0( mod 4) so that h ≡ 4t( mod 12) with t = 0, 1, 2 whence
12r = h − 4t + 12, h − 12r + 12 = 4t and
Gh−12r+12 = G4t = Gt4 .
Further we have now
T h = −G12r+h+2 ∆1−r G−1
14
dj
1−r d j
= −G−t
4 ∆
d log q
d log q
3 1−r−t/3 d j1−t/3
∆
t−3
d log q
3−t −r
d(G
∆
) 3r + t − 3 3−t d∆−r
3
4
+
G
;
=
t − 3 d log q
(3 − t)r 4 d log a
=
hence Ch0 is also the constant term in the Fourier expansion of the function
Vh =
3r + t − 3 3−t d∆−r
G
.
(3 − t)r 4 d log q
In view of the assumption h > 2, we have 3r + t − 3 > 0. The series for G3−t
4
begins with 1 and has again all its coefficients positive. Further, by (12), the
coefficients of the negative powers q−r , . . . , q−1 of q in the derivative of ∆−r
with respect to log q are all negative while the constant term is absent. Hence
the constant term in Vh is negative and ch = ch0 < 0 i.e. ch , 0. The proof is
thus complete.
A most important consequence of the two theorems is the fact that, for every modular form M of weight h, the constant term a0 in its Fourier expansion
is determined by the r Fourier coefficients a1 , . . . , ar that follow, namely by the
formula
a0 c−1
(14)
h (ch1 a1 + · · · + chr ar ).
If, in particular, a1 , . . . , ar are rational integers, then a0 itself is rational and the
denominator of a0 divides ch .
254
BIBLIOGRAPHY
224
As a further remarkable result, we note that the vanishing of a0 follows
from the vanishing of the Fourier coefficients a1 , . . . , ar and hence the vanishing of the modular form M itself, since then the entire modular function W
defined in (9) has a zero at q = 0.
From (10) and (11), it follows that the numbers chl (l = 0, 1, . . . , r) are the
coefficients of q−l in the product of G12r−h+2 with ∆−r as is indeed seen by direct
calculation, which can be slightly simplified by using the formula
∞
Y
m=1
∞
X
(1 − q ) =
(−1)n (2n + 1)qn(n+1)/2
m 3
n=0
Now chr = 1 always. Also we easily obtain the values
c4 = −240 = −24 · 3 · 5;
c6 = 504 = 23 · 32 · 7;
c8 = −480 = −25 · 3 · 5;
c10 = 264 = 23 · 3 · 11;
c12 = −196560 = −24 · 33 · 5 · 7 · 13,
c12,1 = 24 = 23 · 3;
255
c14 = 24 = 23 · 3;
c16 = −146880 = −26 · 33 · 5 · 17,
c16,1 = −216 = −23 · 33 ;
c20 = −39600 = −24 · 32 · 52 · 11,
c20,1 = −456 = −23 · 3 · 19;
c18 = 86184 = 23 · 34 · 7 · 19,
c22 = 14904 = 23 · 34 · 23,
c24 = −52416000 = −29 · 32 · 53 · 7 · 13,
c24,1 = −195660 = −22 · 32 · 5 · 1087,
c26 = 1224 = 23 · 32 · 17,
c18,1 = 528 = 24 · 3 · 11;
c22,1 = 288 = 25 · 32 ;
c24,2 = 48 = 24 · 3;
c26,1 = 48 = 24 · 3.
From this table, we see for example, with h = 12 and h = 24, that the r
numbers chl (l = 0, . . . , r − 1) need not all have the same sign, if h is a multiple
of 4. Such a change of sign always occurs, moreover, for h > 248 since, in
particular, for this h, ch,r−1 > 0 whereas Ch0 < 0. Further, since the number
ch,r−1 is 0 for h = 214 and h = 248, the assertion of Theorem 2 does not hold,
without exception, for all chl (l = 1, 2, . . . , r − 1).
For number-theoretic applications of theorem 1, the prime-factors of ch
are of interest. We get some information regarding these prime factors if in the
BIBLIOGRAPHY
225
statement of Theorem 1 we consider the Fourier coefficients of Gh (h = 4, 6, . . .)
given by (4) instead of the modular form M. Then we have from (14),
X
ch Bh X
=
chl σh−1 (l), σh−1 (l) =
tb−1 (h = 4, 6, . . .)
2h
l=1
t|l
r
(15)
Since the first sum here is an integer, ch must be divisible by the denominaBh
; in any case, the denominator of the Bernoulli numberBh is a divisor
tor of
2h
of ch . By the Staudt-Clausen theorem, however, this denominator is equal to
the product of those primes p, for which p − 1 divides h, Thus, for example, for
h = 24, one sees the reason for the presence of the prime divisors 2, 3, 5, 7, 13
of c24 . On the other hand, for h = 26, the factor 17 of c26 cannot be explained
in this simple manner. By (15), we have however explicitly
2 · 32 · 17 · B26 c26 B26
−
= 24 · 3 · 125 + (125 + 225 ) = 33554481
13
2 · 26
= 3 · 17 · 657931
where thus the factor 17 appears on the right side also and we obtain
B26 =
13 · 657931
2·3
agreeing with the known value.
2 Modular Forms of Hecke
Let K be a totally real algebraic number field of degree g and discriminant d
and let further k, be a natural number which is, inaddition, even if there exists
in K a unit of norm −1. We denote the norm and trace by N and S . Let u be an
ideal, d the different of K and u∗ = (ud)−1 , the complementary ideal to u.
We assign to every one of the g fields conjugate to K, one of the variables
z(1) , . . . , z(g) in the upper half-plane and form with them the Eisenstein series
X′
Fk (u, z) = N(uk )
N((λz + µ)−k )
u|(λ,µ)
where λ, µ run over a complete system of pairs of numbers in the ideal u,
different from 0, 0 and not differing from one another by a factor which is a unit.
The series converges absolutely for k > 2. For the cases k = 1, 2, in order to
256
BIBLIOGRAPHY
226
obtain convergence, the general term of the series is multiplied, as usual, by the
factor N(|λz+µ|−s ), where the real part of the complex variable s is greater than
2 − k and Fk (u, z) is defined as thevalue of the analytic continuation at s = 0.
A comparison with (1) shows that Fk (u, z) gives for g = 1 and k = 4, 6, . . . the
function denoted there as Fk (z). Let hereafter g > 1.
The generalization of (2) gives for the Fourier expansion of Fk (u, z), the
formula
!g
X
(2πi)k
Fk (u, z) = ζ(u, k) +
d1/2−k
σk−1 (u, v)e2πiS (vz) ,
(16)
(k − 1)!
−1
d |v>0
where v runs over all totally positive numbers in d−1 and
X
ζ(u, k) = N(uk )
N(µ−k ),
σk−1 (u, v) =
X
u|(µ)
sign(N(αk ))N((α)ud)k−1 .
(17)
d−1 |(α)u|v
Here the summation is over principal ideals (µ), (α) under the conditions given.
For even k, (17) can be put in the form
X
σk−1 (u, v) =
N(tk−1 )(k = 2, 4, . . .)
(18)
ud∼t|(v)d
where t runs over all integral ideals in the class of ud dividing (v)d. As a
departure from the case g = 1, formula (??) is valid also for k = 2, while for
k = 1, an additional term is to be tagged on. For this reason, it is provisionally
assumed that k > 1.
The function Fk (u, z) is a modular form of Hecke of weight k, since under
the substitutions
αz + β
z→
γz + d
of the Hilbert modular group, it takes the required factor N((γz + d)k ). If we
set all the g independent variables z(1) , . . . , z(g) equal to the same value z in
the upper half-plane, then form Fk (u, z) one obtains an elliptic modular form,
which we denote by Φk (u, z). Its weight is clearly h = kg. Let its Fourier
expansion in powers of q = e2πiz be
Φk (u, z) =
∞
X
n=0
bn qn
257
BIBLIOGRAPHY
227
wherenow the Fourier coefficients b0 , b1 , . . . are determined, in view of (??) by
the formulae
!g
X
(2πi)k
b0 = ζ(u, k), bn =
d1/2−k
σk−1 (u, v)(n = 1, 2, . . .)
(19)
(k − 1)!
S (v)=n
258
Here v runs over all totally positive numbers with trace n in the ideal d−1 and
σk−1 (u, v) is defined by (17). For brevity, let us put
X
σk−1 (u, v) = sn (u, k)(n = 1, 2, . . .);
(20)
S (v)=n
this is a rational integer. Explicitly,
X
sn (u, k) =
sign(N(αk ))N((α)ud)k−1
(21)
∗
u |(α),u|β
αβ>0,S (αβ)=n
where the summation is over the principal ideals (α) and the numbers β in the
field, subject to the given conditions.
On applying (14) with M = Φk (u, z), it now follows
((k − 1)!)g (2πi)−h dk−1/2 ζ(u, k) = −c−1
h
r
X
ch S l (u, k);
(22)
l=1
here we have to take h = kg and r = rh . We note further that N(−1) = (−1)g and
therefore h is always an even number. These formulae permit the calculation
of ζ(u, k), if the ideal u and the natural number k are given. They show, in
particular, that the value
π−kg dk−1/2 ζ(u, k) = ψ(u, k)
is rational and more precisely, its denominator divides ((k − 1)!)g ch . It is to be
stressed here that ch depends only on h = kg but not on the other properties of
the field K.
In the special case g = 2 and thus for a real quadratic field, it was already
known that the number on the left side of (22) is rational and that its denominator divides 4((2k)!)2 v2k where however v still depends on K, being given by
the smallest positive solution y = v of Pell’s equationx2 − dy2 = 4. On the
other hand, with the help of a computing machine, Lang has listed the values
of ψ(u, 2) in 50 different examples for k = g = 2 and remarked, in addition,
that in every one of the cases considered, only a divisor of 15 appears as the
denominator. It is now easy to explain this phenomenon since −2−4 c4 = 15.
259
BIBLIOGRAPHY
228
For every fixed totally real algebraic number field K, we can, following
(3), look upon the positive rational numbers ψ(u, k)(k = 2, 4, . . .) as the analogues of the values 2k−1 (k!)−1 |Bk | formed with the Bernoulli numbers. Unfortunately, however, one cannot expect that for this generalisation, corresponding recursion-formulae and divisibility theorems will exist as for the Bernoulli
numbers themselves. For, they are connected with the properties of the function (e2πz − 1)−1 whose partial fraction decomposition also leads to (3). Indeed,
similar to (??) we have even the simpler formule
!g
X
X
(2πi)k
d−1/2 N(u−1 )
N(vk−1 )e2πiS (vz) (k = 2, 3, . . .);
N((z+µ)−k ) =
(k
−
1)!
∗
u |v>0
u|µ
but in contrast with the case of one variable, the analytic function appearing
here for g > 1 does not have, any more, important properties such as the addition theorem and the differential equation enjoyed by the exponential function.
Further by an important result of Hecke, it is always singular for all real values
of any one of the g variables, so that one cannot as in the case g = 1 obtain the
numbers ζ(u, k) by expansion in powers of z. We get these numbers by going
to the modular forms Fk (u, z) of Hecke, which in spite of their complicated
construction are really connected with one another by algebraic relations.
To the result (22) we may add a remark which relates to a lower bound for
d discovered by Minkowski. The left side of (22) is clearly different from 0 for
even k, since then ζ(u, k) is positive. Consequently among the r sums sl (u, k)
given for l = 1, 2, . . . , r by (21), at least one must be non-empty and hence
there always exist two numbers α, β in K which satisfy the four conditions
u∗ |α, u|β, αβ > 0, S (αβ) ≤ r
with r = rh = rkg . For this assertion, it is optimal to choose k = 2 and hence
h = 2g; accordingly
g
g
r=
+ 1(g . 1( mod 6)), r =
(g ≡ 1( mod 6))
6
6
so that in every case
g
r ≤ + 1.
(23)
6
From the decompositions
(α) = u∗ t, (β) = ub
with integral ideals t, b, it follows that there exist two integral ideals t and b (in
every ideal class and its complementary class), such that
tbd−1 = (v), v > 0, S (v) ≤ r
(24)
260
BIBLIOGRAPHY
229
We therefore have
r
d N(tb) = N(tbd ) = N(v) ≤ (g S (v)) ≤
g
!g
r
N(tb) ≤
d
g
−1
−1
−1
and hence
Min(N(t), N(b)) ≤
r
g
!g/2
g
!g
(25)
√
d
√
Since especially for every real quadratic field, the ideal d = ( d) belongs to
the principal class, the classes of t and b are inverses of each other because of
the first formula (24) and hence the conjugate ideal b′ = (N(b))b−1 is equivalent
to t; further, g = 2 and so r = 1. Since N(b) = N(b′ ), there exists, in every ideal
1√
class of a real quadratic field, an integral ideal of norm ≤
d always.
2
For arbitrary totally real fields of degree g ≥ 2, we always have by (25)
!g
r
1 ≤ N(t) ≤
d,
g
from which, by (23) the inequality
d≥
g g
r
261
6
g
≥ 6g 1 +
!−g
> 6g e−6
(26)
follows. Since in the cases g = 2, 3, 4, 5, 7 the number r = 1, we have then
d ≥ gg
and this is a new result for field degrees 5 and 7. Since e2 > 6, the estimate
gg
d≥
g!
!2
found by Minkowski is better than (??) for all sufficiently large g, but not for
a few small g to which set the given values 5 and 7 belong. For g = 4, it is
well-known that 725 is the smallest discriminant of a totally real field while
44 = 256,
48
7
= 113 + .
9
(4!)2
BIBLIOGRAPHY
230
The special case g = 2 yields also a connection with the reduction theory
of indefinite quadratic forms due to Gauss. Namely, let [κ, λ] be a basis of the
ideal u and hence
√
√
λ b+ d
=
, d = b2 − 4ac, N(u) d = |κλ′ − λκ′ |
κ
2a
with rational integers
a = N(κ)N(u−1 ), b = S (κλ′ )N(u−1 ), c = N(λ)N(u−1 ).
Then the conjugate ideal is u′ = (κ′ , λ′ ) and (κλ′ − λκ′ )−1 [λ′ , −κ′ ] gives the
basis of u∗ = (ud)−1 complementary to [κ, λ]. If we now put
α=
κ′ x2 + λ′ y2
, β = κx1 + λy1
κλ′ − λκ′
withrational integers x1 , y1 , x2 , y2 , then
262
2κκ′ x1 x2 + (κλ′ + λκ′ )(x1 y2 + x2 y1 ) + 2λλ′ y1 y2 x1 y2 − x2 y1
+
2(κλ′ − λκ′ )
2
2ax1 x2 + b(x1 y2 + x2 y1 ) + 2cy1 y2 x1 y2 − x2 y1
+
=
√
2
2 sign(κλ′ − λκ′ ) d
αβ =
and the two conditions
S (αβ) = 1, αβ > 0
go over into
x1 y2 − x2 y1 = 1, |2ax1 x2 + b(x1 y2 + x2 y1 ) + 2cy1 y2 | <
√
d.
But this precisely means that the binary quadratic form au2 + buv + cv2 of
discriminant d goes over, under the linear substitution u → x1 u+x2 v, v → y1 u+
y2 v of determinant
√ 1, into a properly equivalent form whose middle coefficient
is smaller than d in absolute value. This means again that in the transformed
form both the outer coefficients are of opposite signs and this is just the essence
of reduction theory.
We have still to consider the case k = 1 excluded hitherto. In this case [3],
in contrast to (??), the constant term is
b0 = ζ(u, 1) + ζ(u∗ , 1)
where we have to define
ζ(u, 1) = N(u) lim
s→0
X
u|(µ)
N(µ−1 )N(|µ|−s )
BIBLIOGRAPHY
231
and also change (22) correspondingly. This way we do not obtain the individual
numbers ζ(u, 1) and ζ(u∗ , 1) but only their sum. By the way, this is always 0 if
either g = 2 or u2 d is a principal ideal (γ) and N(γ) < 0.
3 Examples
In conclusion, we give three simple examples for evaluating ζ(u, k) using(22),
where the result can also be checked in another way.
With k = g = 2, we have h = 4, r = 1 and
N(u2 )
X
u|(µ)
N(µ−2 ) = −
(2π)4
π4
s
(u,
2)
=
√ s1 (u, 2);
1
c4 d3/2
15d d
here sn (u, k) for k = 2, 4, . . . and n = 1, 2, . . . is given by (18) and (20). If
a+δ
we take, in particular, d = 4.79, u = b = (1), then d = (2δ), v =
with
2δ
√
2
δ = 79 and a = 0, ±1, . . . , ±8, −N(a + δ) = 79, 2 · 3 · 13, 3 · 5 , 2 · 5 · 7,
32 · 7, 2 · 33 , 43, 2 · 3 · 5, 3 · 5. We now note that 2 = N(9 + δ), while the cube of
the prime ideal p = (3, 1 + δ) gives a principal ideal. For the determination of
the principal ideal divisors (τ) of (a + δ), we have to consider besides the trivial
decomposition (a + δ) = (1) · (a + δ), only
a + δ
(a + δ) = (9 + δ)
9+δ
with odd a; hence, for
σ1 (o, v) = o(a) =
X
N((τ)),
o|(τ)|(a+δ)
we have the values
σ(0) = 1 + 79 = 80,
σ(±1) = 1 + 78 + 2 + 39 = 120,
σ(±2) = 1 + 75 = 76,
σ(±4) = 1 + 63 = 64,
σ(±3) = 1 + 70 + 2 + 35 = 108,
σ(±5) = 1 + 54 + 2 + 27 = 84,
σ(±6) = 1 + 43 = 44,
σ(±7) = 1 + 30 + 2 + 15 = 48,
σ(±8) = 1 + 15 = 16.
Accordingly, in the present case,
s1 (o, 2) = 80 + 2(120 + 76 + 108 + 64 + 84 + 44 + 48 + 16) = 1200
263
BIBLIOGRAPHY
232
X
N(µ−2 ) =
o|(µ)
10π4
√
79 79
inagreement with the value found by me elsewhere. For u = p = (3, 1 + δ), we
obtain correspondingly
s1 (p, 2) = 0 + (3 + 6 + 13 + 26) + (3 + 5 + 15 + 25)+
+ (5 + 10 + 7 + 14) + (3 + 7 + 21 + 9)+
+ (3 + 6 + 9 + 18) + 0 + (3 + 6 + 5 + 10) + (3 + 5) = 240
X
2π4
N(p2 )
N(µ−2 ) =
√
79 79
p|(µ)
and the same for u = p′ = (3, 1 − δ). Since the 3 ideal classes of the field
are represented by 1, p, p′ , we have therefore for the zeta function ζK (s) of the
field,
π4
14π4
ζK (2) =
√ (10 + 2 + 2) =
√ ,
79 79
79 79
while from the law of decomposition, we get, likewise, on using the formula
!
!
d
X d!
l
(2πi)k X d
−k
n =−
pk
,
√
n
d
1 · k! d l=1 l
n=1
!
k
X
k
Pk (x) =
Bl xk−l (k = 2, 4, . . .)
l
l=0
that
!
∞
X
316 −2
ζK (2) = ζ(2)
n
n
n=1
!
!
316
l
π2 (2π)2 X 316
P2
√
6 2 · 2! 316 l=1 l
316
!
!
39
2l − 1
14π4
π4 1 X 79
1−
=
=
√
√ .
79
12 79 2 l=1 2l − 1
79 79
=
We now take k = 12, g = 2; then h = 24, r = 3 and
X
N(u12 )
N(µ−12 )
u|(µ)
264
BIBLIOGRAPHY
233
265
24
(2π)
(c24,1 s1 (u, 12) + c24,2 s2 (u, 12) + s3 (u, 12))
(11!)2 c24 d23/2
π24
=
√ (−195660s1 (u, 12) + 48s2 (u, 12) + s3 (u, 12)).
2 · 310 · 57 · 73 · 112 · 13d11 d
√
In the special case d = 5, there exists only one class d = ( 5) and the totally
√
±1 + 5
−1
positive numbers v of the ideal d with trace 1, 2, 3 are given by
√ ,
2 5
√
√
√
√
a+3 5
±1 + 5
a+ 5
, ±2 + 5
√ (a = 0, ±1, ±2),
√ (a + ±1, ±3, ±5). Since
2
5
2 5
√
√
√
√ ±1 + 3 5 ±3 + 3 5 ±5 + 3 5
√
are units and also 5, ±1 + 5,
,
,
are inde2
2
2
composable, we have
=−
s1 (o, 12) = 2 · 111 = 2, s2 (b, 12) = 111 + 511 + 2(111 + 411 ) + 2 · 111
= 57216738,
s3 (o, 12) = 2(111 + 1111 ) + 2(111 + 911 ) + 2(111 + 511 ) = 633483116696
and thus
X
N(µ−12 ) =
o|(µ)|
24 · 691 · 110921π24
√
310 · 516 · 73 · 112 · 13 5
On the other hand, in the present case too, we have
X
N(µ−12 ) = ζK (12)
o|(µ)
!
!
!
4
∞
X
l
(2π)24 B12 X 5
5 −12
n =
P12
= ζ(12)
√
n
5
(2 · 12!)2 5 l=1 l
n=1
with the same result, where this time the factor 691 comes in through the numerator of B12 .
Finallylet k = 2, g = 3 so that h = 6, r = 1 and
N(u2 )
X
u|(µ)
N(µ−2 ) =
(2π)6
8π6
s1 (u, 2) =
√ s1 (u, 2).
3/2
c6 d
63d d
In particular, let d = 49 and hence K, the totally real cubic subfield of the field
of 7th roots of unity. It is generated by
ρ = 2 − ǫ − ǫ −1 = (1 − ǫ)(1 − ǫ −1 )
266
BIBLIOGRAPHY
234
with ǫ = e2πi/7 and has class number 1, as is also evident from (25). Since
[1, ρ, ρ2 ] is a basis of o and N(ρ) = 7, we have (ρ3 ) = 7 and so d = (ρ2 ). We
have hence to determine for
v = x + yρ−1 + zρ−2 ,
the solutions of S (v) = 1, v > 0 with rational integers x, y, z. A short computation gives
ρ3 = 7(ρ − 1)2 , S (ρ) = 7, S (ρ2 ) = 21, S (ρ−1 ) = 2, S (ρ−2 ) = 2,
3
S (ρ−3 ) = 2 + ,
7
S (v) = 3x + 2y + 2z = 1
By the substitution x = 1 + 2u with integral u, it follows that y = −1 − y − 3u
and the condition v > 0, and consequently ρ2 v > 0, goes over into
(2ρ2 − 3ρ)u + (1 − ρ)z + ρ2 − ρ > 0.
(27)
If we set again
ǫ k + ǫ −k = 2 cos
2kπ
= λk
7
(k = 1, 2, 3),
then
π
3
π √
< λ1 < 2 cos = 2 < ,
3
4
2
1
π
π
π
− < − < −2 sin
= λ2 < 2 cos = 0
2
7
14
2
√
3
π
π
− 2 = 2 cos π < λ3 = −2 cos < −2 cos = − 3 < − .
7
6
2
1 = 2 cos
267
With this, we obtain from (27) by a simple modification, the three conditions
z + λ1 − 2 > (λ2 + 5)u, z + λ2 − 2 < (λ3 + 5)u,
z + λ3 − 2 < (λ1 + 5)u
whence, it follows, in particular, that
λ3 < u < λ1
BIBLIOGRAPHY
235
Therefore, for u, only the values 0, ±1 have to be considered. Of these however,
u = 1 leads to the contradiction
5 < λ2 − λ1 + 7 < z < λ3 − λ2 + 7 < 6.
For u = 0, we have
2 − λ1 < z < 2 − λ2 < 2 − λ3 ,
and hence either z = 1, y = −2, x = 1 or z = 2, y = −3, x = 1 and for u = −1,
λ3 − 2 < z < λ2 − 2 < λ1 − 2
and hence z = −3, y = 5, x = −1. The first of the three solutions formed above,
gives
ρ
ρ2 v = (ρ − 1)2 , v = .
7
and the other two v must therefore be conjugate to this, as we can also check
directly. In all the three cases, ρ2 v proves to be a unit and hence (v)d = o, t = o,
s1 (o, 2) = 3 and
X
N(µ−2 ) =
o|(µ)
23 · π6
8π6 · 3
.
√ =
3 · 74
63 · 49 · 49
In order to verify this result in another way, we note from the law of decomposition for the cyclotomic field, that
X
o|(µ)
N(µ−2 ) = ζK (2) = (1 − 7−2 )−1
2
Y
(w1 + ηl w2 + η−l w3 )
l=0
2
=
with
wk =
7
(w3 + w32 + w33 − 3w1 w2 w3 )
24 · 3 1
∞
X
268
−2
2πi/3
(7n + k) (k = 1, 2, 3), η = e
.
n=−∞
But now



wk = 

2

!2
π 
2π
(2 − ǫ k − ǫ −k )−1 =
 =
k 
7
7 sin
7
!6
!6
2π
2π
S (ρ−3 ) =
2+
w31 + w32 + w33 =
7
7
2π
7
3
7
!
!2
ρ−1
BIBLIOGRAPHY
236
2π
w1 w2 w3 =
7
!6
2π
N(ρ ) =
7
−1
!6
·
!6
·2=
1
7
and indeed we have again
X
o|(µ)
N(µ−2 ) =
72 2π
4
2 ·3 7
23 · π6
3 · 74
Bibliography
[1] Meyer, C: Über die Bildung von elementar-arithmetischen Klasseninvarianten in reell-quadratischen Zahlkörpern Berichte. math. Forsch-Ist.
Oberwalfach Z, Mannheim 1966, 165-215.
Lang, H: Über eine Gattung elementar-arithmetischer Klasseninvarianten
reell-quadratischer Zahkkórper. Inaug Diss. Köln. 1967 VII u. 84 S.
Siegel, C. L.: Bernoullische Polynome und quadratische Zahlkörper.
Nachr. Akad. Wiss. Gottigen, Math-Phys. K1. 1968, 7-38.
Barner, K: Über die Werte der Ringklassen-L-Funktionen reellquadratischer Zahlkörper an natürlichen Argumentstellen. J. Numbr. Th.
1. 28-64 (1969).
[2] Klingen, H: Über die Werte der Dedekindschen Zetafunktionen, Math.
Ann. 145, 265-272 (1962).
[3] Gundlach, K. B: Poincarésche und Eisensteinsche Reihen zur
Hilbertschen Modulgruppe. Math. Zeitschr. 64, 339-352 (1956); insbes
350-351.
237