Download non-abelian classfields over function fields in special cases

Actes, Congrès intern, math., 1970. Tome 1, p. 381 à 389.
NON-ABELIAN CLASSFIELDS
OVER FUNCTION FIELDS IN SPECIAL CASES
by YASUTAKA IHARA
1. General formulations and conjectures.
1.1. Primes and Conjugacy Classes Principle.
One of our basic ideas is that a certain type of infinite discrete groups T plays a
central role in arithmetic of non-abelian extensions of algebraic function fields of one
variable over finite fields (abbrev. function fields). An origin of this idea was the
following question: is there any identity between the Riemann ^-function of an arithmetic field K and a " Selberg type ^-function " of a discrete group T, in some cases?
Indeed, if we assume such an identity for a function field K, then we can proceed and
meet, in a fairly natural way, the concept of " primes and conjugacy classes principle "
(abbrev. p. & c.c. principle), and then of " non-abelian classfield theory of T-type ". We
shall begin with this explanation. Another origin was an observation that if we
introduce such a group as T = PSL2(Zip)), where p is a prime number and
Z « = {fl/p>|ûWEZ},
then this theory holds for such a group [5]. But this explanation will be left to § 2.
Recall that the original Selberg C-function is defined with respect to a discrete subgroup T of G = PSL2(R) with finite volume quotient. While the Riemann C-function
describes the distribution of prime divisors of an arithmetic field, the Selberg C-function describes that of T-conjugacy classes in the space of G-conjugacy classes. By
" Selberg type Ç-functions, " we vaguely mean some " ^-functions " connected with
the distribution problems of T-conjugacy classes, where G is some more general topological group, say, of Lie type. Since the space of G-conjugacy classes of a Lie type
group G is roughly identical to the disjoint union of mutually non-conjugate tori T
of G, we may fix one torus T in considering such a " C-function ".
Let us formulate the above question in a more explicit way. Take an infinite discrete
subgroup T of a topological group G, and let T be a closed abelian subgroup of G.
Call T* the set of all such element r G T that the centralizer of t i n G coincides with T.
Then g~1tg = t' (t, t' e T*, g e G ) implies g~lTg = T. Now suppose that T contains
an open compact subgroup T0 with T/T0 ^ Z (1). For each t E T, call deg (t) (the degree
of t) the absolute value of the image of t by the induced homomorphism T -* Z.
Since T0 is the unique maximal compact subgroup of T, it is clear that deg (t) is inva(*) This assumption is natural, when we presuppose an identity between a Selberg type
C-function w.r.t (G, T, T) and a congruence (-function.
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Y. IHARA
B4
riant by any topological automorphism of T. In particular, if t, t' E T* are G-conjugate and hence conjugate by the normalizer of T, then deg (t) = deg (f). Assume
now that if some element ^ 1 of T is G-conjugate to some t E T, then t E T*. Call
a subset S of F, T-bolic, if there is some g e G with g~1Sg cz T. It is easy to see that a
subgroup H of T is maximal T-bolic if and only if either of the following two equivalent
conditions (i), (ii) is satisfied: (i) H is the centralizer of some T-bolic element ^ 1 of T ;
(ii) for every y G H with y ^ 1, y is T-bolic and H coincides with its centralizer. It
is clear that if H, H' are two distinct maximal T-bolic subgroups, then H n H' = { 1 }.
Now let Jti? denote the set of all infinite (2) maximal T-bolic subgroups of T. Then if
HEjtf, its torsion subgroup H° is finite and the quotient H = H/H° is isomorphic
to Z. Suppose that we can assign one choice of isomorphism H = Z for each HE#?9
in such a way as to be compatible with the conjugations by elements of T (3). Let T
act on JP by the conjugation. From each T-conjugacy class of subgroups in Jf, choose
a representative H9 and for each representative H9 let W denote the positive generator
of H; i. e., the generator of H that corresponds to 1 by the above chosen isomorphism
H ^ Z. Thus, H, % and its T-conjugacy class { % } r (which is more intrinsic than H, n)
are in one-to-one correspondence with each other. Note that if T is torsion-free,
{ n }r * runs over all primitive T-bolic T-conjugacy classes; i. e., the T-conjugacy classes
of such T-bolic elements of T that generate some HEJ?.
NOW, define the degree
deg {7f } r of { K } r , by deg { % } r = deg (r), where t = g - 1 ^ G T and % E H is a representative modulo H° of 7c. By the above remarks, it is well-defined, and is a positive
integer. Now define the C-function by
Cr(u) = ri(i-" d e g { Ä } r )- 1 .
This is an analogue of Selberg C-function in idea, but in form, it is an analogue of congruence C-functions (4) (5).
Now we can raise our question more concretely: is there any (G, T, T) of the above
type and a function field K, such that Cr(u) essentially coincides with the congruence
C-function of K, at least for the principal parts? Since the C-function of K is by definition H ( l ~ wdegP)~ \ (P: prime divisors of K,) this question is refined to the following:
p
Is there any (G, T, T) and K, and a " natural " degree-preserving one-to-one correspondence between a set p(T) of almost all {n}r and a set p(K) of almost all prime divisors
of K?
If there is such a correspondence, we shall say that the primes and conjugacy classes
principle holds between p(K) and p(T). That this principle actually holds for some K
(2) If G/r is compact, then any maximal T-bolic subgroup H is infinite, since if g lHg c T,
then T/g~iHg is compact.
(3) This is possible if and only if the mutually inverse elements of H are not T-conjugate
to each other for any H ; and when possible, in many ways so. But the " reciprocity ", i. e.,
(ii) (b) of § 1.2, can be expected only for good choices of isomorphisms. Thus, our conjectures
and results that follow depend on their good choices.
(4) If G/r is compact, or if (G, T, T) belongs to the type that we consider in § 1.3, there are
at most finitely many {7f } r with the given degree, and hence Çr(u) is well-defined as a formal
power series.
(5) Note that Cr(") does not depend on the choice of the isomorphisms H ^ Z.
NON-ABELIAN CLASSFIELDS OVER FUNCTION FIELDS IN SPECIAL CASES
and T, is a basic point in making our program and proving some results on "
abelian classfield of T-type ". Indeed, then F would play the role of the ideal
group (in the sense of Takagi) of abelian classfield theory, and the conjugacy
{n } r E p(F) corresponding to a prime divisor P of K would play the role of the
class determined by P.
383
nonclass
class
ideal
1.2. Abstract formulation of non-abelian classfield theory of T-type. We assume
that the p. & c. c. principle holds between some p(K) and p(r);
P(r)3{7f}r tf ^ p ( K ) .
We shall say that a non-abelian classfield theory of T-type is valid between K and F
if there is an infinite Galois extension ft of K and an injective isomorphism i of F
into the Galois group g = Gal (ft/K), satisfying the following conditions (i), (ii):
(i) i(F) is dense in g. The subgroups Y'aYof finite indices and the closed subgroups g' <= g of finite indices are i?i a natural one-to-one correspondence.
Hence F' also correspond in a one-to-one manner with finite extensions K' of K
contained in ft. The next condition (ii) comes out naturally from the idea that the
p. & c. c. principle should be assumed for all corresponding K' and F', in a compatible
way.
(ii) Let P E p(K) and let { n } r G p(T) be the corresponding F-conjugacy class.
there is a prime factor p of P in ft such that
Then,
(a) i(H°), resp. the topological closure of i(H) in g, are the inertia group, resp. the
decomposition group, of p in R/K;
(b) i(n) is the Frobenius automorphism of p in R/K.
This condition (ii) would describe the law of decompositions of prime divisors P
of p(K) in ft/K completely, and would imply that the p. & c. c. principle holds for all
corresponding K' and T" in a compatible way. In fact, it is enough to connect
iGOPhr
with
yHy~1nr
for each yET and P G p(K). It is clear by definition that if the classfield theory of
T-type is valid between K and T, then it is also valid for all corresponding pairs K'
and r.
1.3. Main conjectures. The detailed studies of some selected cases gave us a strong
hope that the classfield theory of T-type is actually valid for the groups F of the type
defined below, and tempted us to propose a following series of conjectures (C 1) ~ (C 5).
It is stronger than the classfield theory of T-type defined in § 1.2. So far, it is proved
only for some special cases of F, but seems to be supported also by some other results
on T (e. g., § 3.1),
We shall specify (G, T, T) as follows:
G = PSL2(R) x PSL2(k?)
(topological group),
where R and kp are the real and a p-adic field respectively, and PSL2 = SL2/± 1.
For each subset S <= G, SR resp. Sv denote its projection to GR = PSL2(R) resp.
Gp = PSL2(/cp). Now, T is a discrete subgroup of G with finite volume quotient G/r,
which is essentially indecomposable, in the sense that FR resp. Fv are dense in GR resp. Gv.
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Y. IHARA
B4
For simplicity's sake, we shall assume F to be torsion-free. We shall take
T = PS02(R) x (the diagonals),
so that T/T0 s Z with T0 = PS02(R) x (unit diagonals). It is easy to check all
assumptions of § 1.1 on (G, F, T). Let p(F) be the set of all { n } r . It is an infinite
set, and the C-function can be explicitly calculated (§ 3.1). Besides p(F), a certain
finite set p^F), and the degrees of its elements are defined ([7], vol. 2, Chap. 1). p^ÇF)
is empty if and only if G/F is compact. Put F° = F n(GRx
V), where V= PSL2((9^)
and (9V is the ring of integers of kp. Then its projection FR is a discrete subgroup
of GR with finite volume quotient. Let g denote the genus of FR, and let s be the number
of distinct cusps in a fundamental domain of FR. Then g — 1 + s/2 > 0, and
s=
£
deg P.
Pep 0 0 (D
Put q = Np9 and H = (q — l)(g — 1 + s/2). Then H is always a positive integer
([7], vol. 2, Chap. 1). Now, our main conjectures are the following (CI) ~ ( C 5 ) :
(C 1) Each F defines a function field K with genus g and with exact constant field Fq2,
and the p. & c. c. principle holds between p(K) and p(F). More precisely, the set of
all prime divisors of K is decomposed into three mutually disjoint subsets p(K)9 p^K)
and S(K); and we have degree-preserving one-to-one correspondences
V(K)
~ p(r),
pjK) ~ p j r ) ,
which agree with (C 2) ~ (C 5).
We shall call the prime divisors of p(K), p^K)
special, respectively.
and <5(K) ordinary, cuspidal, and
(C 2) There are exactly H special prime divisors, and they are of degree one over Fq2.
Moreover, there is a differential co of K of degree (q — l)/2 (resp. q — 1, if 2 \ a), whose
divisor (co) equals
w
=
{ Up}{
FI Ö}~ ( *" 1 ) / 2
(resp. W2).
(C 3) Non-abelian classfield theory of F-type is valid between K and F (for the above
p(K) ~ p(F)).
(C 4) (i) The ordinary and the special prime divisors are unramified in ft; (ii) the special
prime divisors are decomposed completely in ft (6); (iii) the cuspidal prime divisors are
at most tamely ramified in ft; (iv) the inertia group and the Frobenius of cuspidal prime
divisors in ft can also be described in the language of p^F) (7).
(C 5) ft is the maximum Galois extension of K satisfying the conditions (i) (ii) (iii)
of (C4).
Note. — Our conjectures implicitly contain the following.
Let GR = PSL2(R)
act on the complex upper half plane S by T -> (ax + b)(cz + d)~l I I
j EGR\ .
(6) This is the very reason why the special prime divisors do not correspond to any T-conjugacy classes (the Frobenius conjugacy class is trivial!).
(7) We omit this detail here. It can be formulated easily by referring to [7], vol. 2 (Chap. 1,
Th. 3 and Chap. 5, Th. 5).
NON-ABELIAN CLASSFIELDS OVER FUNCTION FIELDS IN SPECIAL CASES
385
Let HE#£.
Then since 7^ = PS02(R)9 HR has a common fixed point z on £. Let P
be any subgroup of F with finite index and put A' = Ff n (GR x V), so that AR is a
fuchsian group. Then, we conjecture that there is a suitable algebraico-geometric
model <$u. of A'R\£>, such that the reduction of #A< modulo some extension of p gives
a model of iC' (of § 1.2 (/)), and that the reduction of the point z gives the element of
p(K') corresponding to Hf = H n P G p(T'). In the special cases where our conjectures are solved, the compatible p. & c. c. principle is established in this way.
2. Solved cases.
2.1. Elliptic modular case.
Let p be a prime number, and consider the ring Z (p) = {a/px \a9 tE Z}. Let F
be a subgroup of F(\) = PSL2(Z{p)) with finite index. Then by the diagonal embedding T can be considered as a discrete subgroup of G = PSL2(R) x PSL2(Qp) (Qp: the
p-adic field) with finite volume quotient, and with dense image of projection in each
component of G.
THEOREM. — Let F be as above, and moreover torsion-free. Then F satisfies
(C 1) ~ (C 4). The torsion-freeness assumption can be dropped if we modify (C 2) (C 4)
in a suitable way. (In particular, the non-abelian classfield theory of F-type is valid
for these groups).
The proof is given in [7], vol. 1, 2; Chap. 5. To prove them by our method is equivalent to synthesizing and reconstructing carefully in the language of the group T(l)
the various results, on complex multiplication theory of elliptic curves (Deuring [1]),
and on modernized and generalized Kroneckerian type theory of elliptic modular
functions (Shimura [14] for char. 0, Igusa [4] for char, p) (8). The congruence subgroup
property of F(i) proved by Mennicke [10], Serre [13] is also used. See also § 4. As an
example, take F = T(2) (the principal congruence subgroup of level 2; p ^ 2). It
is torsion-free. The corresponding K is rational; K = Fp2(x). Identify the prime
divisor P of K with the residue class of x, and write P = Pa for x = a (mod P). Then,
pjK) ={P0,P1,Poo},
and <5(K) = {P a | u(a) = 0 } ; where u(x) is a polynomial
of degree (p — l)/2 defined by
u(x) = E?=0[
I xl
(r =(p-
l)/2) (9).
The p. & c. c. principle p(K) <-> p(F) is established by the process described in the above
note (§ 1.3). Namely, let AÄ = r £ = the principal congruence subgroup of PSL2(Z)
of level 2. Let À(z) be the ^-function; i. e., a generator of the field of automorphic
functions w. r. t. AR, whose values at the three cusps are 0, 1, oo. Then
p(F) 3{n}r
-» H -• z -• k0 = X(z) mod Sß -• P = PAo
(8) IGUSA [4] is directly used in constructing ft and proving (C 4) (i), (Hi).
(9) By HASSE-DEURING (cf. [1]), u(a) = 0 if and only if Y2 = X(X - 1)(X - a) is a super-
singular elliptic curve. That a e Fpi and a ^ 0, 1 follows from this (or also directly, by using
our definition of œ given in [9]). That all roots a of u(x) = 0 are simple was directly proved
by IGUSA [3], I am grateful to IGUSA and DWORK, since I was much inspired by [3] and
DWORK [2] (§ on elliptic curves).
I -
13
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Y. IHARA
B4
defines the bijection { ïï } r <-» P of p(F) «-> p(K). Here ty is a fixed prime factor
of p (10). The field ft is obtained from the composite of Igusa's fields of modular
functions for all levels # 0 (mod p), by lowering the field of constant down to Fp2
in a suitable way. Once this is proved, the ramification properties (i. e., the prime
divisors P ^ P 0 , Pl9 P œ of K are unramified and P = P0,P1,Pao are at most tamely
ramified in ft) are reduced to the Igusa's theorem [4]. Moreover, by our theorem,
P G <5(K) are decomposed completely in ft. But we do not know whether ft is
characterized by these ramifications and decompositions properties. For example,
S(K) = { P _ 1 } f o r p = 3; hence (C 5) is to conjecture that ft is the maximum Galois
extension of K in which P ^ P0,P±,Pm are unramified, P 0 , Pl9 P«, are at most tamely
ramified, and P _ ! is decomposed completely.
The differential co of (C 2) is quite an interesting one. It is given by
{x(l-x)YP~1)/2
for the above example.
(dxf
We can prove that:
THEOREM. — co is invariant, up to the signs, by all separable modular transforms x -> x'
(cf. [8]).
Here, x -*• x' is called a separable modular transform if the elliptic curve
Y2 = X(X - 1)(,Y - x')
is separably isogenous to Y2 = X(X — 1)(X — x). The above theorem is also equivalent to that œ is invariant, up to the signs, by all automorphisms of $t/Fp2. Conversely,
a differential r\ ^ 0 of ft (of higher degree) having this invariance property must have
the form: rj = c-coh; CEFP , /IGZ, h > 0. There is no analogue of co in characteristic 0. Another theorem on co is the following. Take a (p — l)/2-th root cox of co
in a separable extension of K. Then:
THEOREM. — (i) co^ is invariant by the Cartier operator, (ii) Let z = z(X) be the
inverse of X(z), and let ( # ) be the Schworf equation for dz:
(#)
2(dz/dX)(d3z/dX3) - 3(d2z/dX2)2 _ X2 - X + 1
(dz/dX)2
" X2(l - X)2 '
Replace X by x and consider ( # ) in characteristic p. Then it is satisfied by co1 in place
of dz. (iii) cox is uniquely characterized by (i), (ii) up to Fp -multiples.
The differential co can be defined in a very natural way in more general cases. This,
and the proof of the theorem (in generalized form) are given in [9].
Note. — The differential dz of the inverse of automorphic functions is always transcendental (unlike the elliptic functions case, where dz is the invariant differential on
the elliptic curve). Nevertheless, [9] shows that in certain cases there is a natural
algebraic differential col in characteristic p which plays a role of " dz (mod p) ".
(10) The choice of one isomorphism H ^ Z for each H, and the injection i : F -> g of
§1.2 are also defined w. r. t afixedprime factor ty of p. We must take the same Sß to validate
our theorem. The effect of changing Sß is of subtle nature ([7], vol. 2 ; Chap. 5).
NON-ABELIAN CLASSFIELDS OVER FUNCTION FIELDS IN SPECIAL CASES
387
2.2. Some quaternion modular cases.
Other known examples of F are obtained from some quaternion algebras B. If F
is the center of 5, F is defined w. r. t. an order ® of B and a prime divisor S$ of F (cf. [7],
vol. 1, Chap. 4). In these cases, FR is the unit group of 0 , and hence belongs to the
fuchsian groups of Poincaré-Fricke type. For these fuchsian groups, Shimura [15] [16]
proved beautiful arithmetic properties of the quotient FR\$>. His theories, combined
with some detailed studies of endomorphism rings of abelian varieties (esp. their
behavior under the reduction processes), may give us enough tool for proving some of
our conjectures. Partial results along this line were obtained by Shimura and by
(our student) Morita:
THEOREM (Shimura) ( n ). — For almost all prime numbers p that remain träge in F
(i. e.9 Sß — p)9 F satisfies the p. & c. c, principle in the way explained in the note (§ 1.3).
Moreover, the number of special prime divisors can be computed, which agrees with (C 2).
If F = Q, this result would imply the main parts of (C1) ~ (C 4), but for almost
all p. Recently, Y. Morita [11] claimed:
THEOREM (Morita). — If F = Q, the main parts of (C 1) ~ (C 4) are valid for all p
not dividing the discriminant of B.
His proof is based on Shimura's and Mumford's theory of moduli of abelian varieties,
Tate's result on endomorphisms of abelian varieties, and on our theorem on Cr(M)
(immediately below) (12).
3. Some related results.
3.1. The C-function of F.
Here we sketch the results of our computations of Cr(M)> where (G, F, T) is as in § 1.3.
n a - Piu)(\ - p\u)
13
THEOREM ( ). -
««) x
\[
(1 - u ^ T
1
= ^
^
ry-
* (* - ")H>
with pip! = q2m9 \ Pi\9 \ p't\ <> q2m, Pi, PÏ ¥" 1, q2- H is a positive integer defined in § 1.3.
If T is not assumed torsion-free, then the formula is somewhat more complicated.
In any case, this result agrees with ( C l ) (C2).
( n ) This was informed by Shimura's letter to the speaker dated May, 1968. He announced
a somewhat stronger result, but it cannot be explained briefly.
(12) Thus, it is very long and involved. We may say that an interest of the (partial) proof
of our conjectures in quaternion modular cases along this line lies more on the corelation
between some problems on abelian varieties and our problems (see also § 4).
(13) This has been announced in [5] in the case G/F: compact, F : torsion-free, and proof
for this case is given in [7], vol. 1, Chap. 1. The general case is proved in [7], vol. 2, Chap. 1.
It is on one hand based on Eichler-Selberg trace formula but on the other hand, it requires a
detailed study of T-bolic and parabolic elements of F. Labesse then offered an alternative
proof in the G/T : compact, F : torsion-free case, by using L2(G/F) in place of Eichler-Selberg
(a letter to the speaker; Oct. 1969).
388
Y. IHARA
Note. — We can express f j ( l — p{u){l — p\u) by some Hecke polynomial.
B4
By
i-l
combining this with Shimura's work [16], it can be shown that the first term on the
right side is a congruence C-function for almost all Sß, in quaternion modular cases.
But this is weaker than the " p. & c. c. principle ", since this still does not guarantee
any natural and compatible one-to-one correspondence between p(K) and p(F).
3.2. The Gp-fields.
The groups F of § 1.3 correspond in a one-to-one manner with the Gp-fields over C
([7]), vol. 1, Chap. 2). Roughly speaking, Gp-field is an algebraic function field having
non-compact automorphism group Gp = PSL2(kp) and satisfying some conditions on
ramifications. One basic theorem is that we can lower the field of constant of Gp-fields
down to algebraic number fields, and that under a certain (not too restrictive) condition
on T, it can be done in a unique way ([7], vol. 1, Chap. 2, Pt. 2). This proof is long,
but uses only some group theories of Gp and a deformation theory of F given in [7].
In our proof, a basic point is that Gp-fields have sufficiently many automorphisms (14).
The relation between the Gp-fields and our problems (which seems essential) is roughly
explained in [6], [7].
4. Concluding remarks.
Our knowledge on non-abelian mathematics is so narrow that we could only have
touched a part of " an iceberg above water ". Here we are satisfied by giving a concrete
program for " non-abelian classfield theory of T-type " (which seems fairly probable),
solving some special cases, and by showing that the problem is closely related to other
arithmetical problems. We shall conclude this talk by the following remark. Function field is often compared with algebraic number field. The former has analogous
but simpler structure than the latter, and also allows geometric treatments. Thus,
some problems that offer no clues at all for number fields may offer some for function
fields. This was first shown by Weil's proof of the Riemann hypothesis for function
fields. So far, it does not help prove the Riemann hypothesis for number fields, but
the related works of Hasse, Weil, etc. have shown that (some) arithmetic problems
on function fields are closely related to some other problems (esp. complex multiplication theory) on number fields, not by a formal analogy between two problems, but more
closely, by a " relation between warps and wooves of the same (if not apparently so)
problem ". Now, a relation of our problem with complex multiplication theory is
also of this sort. We have been talking in a view-point of trying some non-abelian
mathematics on function fields (which are too difficult for number fields), but from
another view-point, it is a " woof " of complex multiplication theory. Namely, if
we call a " warp " of complex multiplication theory that theory of " fixed imaginary
quadratic lattice and variable p ", then a woof is that theory of " fixed p and variable
imaginary quadratic lattices ". There is an admirable generalization of warps (of
complex multiplication theory) by Shimura [16], but its wooves are by no means
(14) Usefulness of this point is also stressed in a recent work of PJATEZKII-SAPIRO [12].
NON-ABELIAN CLASSFIELDS OVER FUNCTION FIELDS IN SPECIAL CASES
389
complete (only touched), and these are almost equivalent to our problems (C 1) ~ (C 4)
for the quaternion modular groups! Finally, we note that the use of the groups F
(of type § 1.3, § 2) is very natural and convenient for " woof theories ".
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functions, J. Math. Soc. Japan, 13 (1961), pp. 275-331.
—. — Construction of class fields and zeta functions of algebraic curves, Ann. of
Math., 85-1 (1967), pp. 58-159.
—. — Local representations of Galois groups, Ibid., 89-1 (1969), pp. 99-124.
—. — On canonical models of arithmetic quotients of bounded symmetric domains,
Ibid., 91-1 (1970), pp. 144-222.
University of Tokyo
Department of Mathematics,
Hongo, Bunkyo-ku,
Tokyo (Japon)