Download MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS 1

MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS
DOHOON CHOI, SUBONG LIM, AND ROBERT C. RHOADES
Abstract. In his last letter to Hardy, Ramanujan described ‘mock theta functions’. For each of his
examples f (q), Ramanujan claimed that there is a collection {Gj } of modular forms such that for each
root of unity ζ, there is j such that
lim (f (q) − Gj (q)) = O(1).
q→ζ
Moreover, Ramanujan claimed this collection must have size larger than one. In his 2001 PhD thesis,
Zwegers showed that the mock theta functions are the holomorphic part of a harmonic Maass form. We
prove there must exist such a collection by establishing a more general result for all holomorphic parts of
harmonic Maass forms. This complements the result of Griffin, Ono, and Rolen that shows such a collection
cannot have size one. These results arise within in the context of Zagier’s theory of “quantum modular
forms”. A linear injective map is given from the space of mock modular forms to quantum modular forms.
Additionally, we provide expressions for “Ramanujan’s radial limits” as L-values.
1. Introduction
In his 1920 letter to Hardy [1], Ramanujan constructed seventeen examples which he called “mock theta
functions”. Ramanujan claimed that these functions satisfiy: (1) they had infinitely many singularities
at rationals, (2) it was impossible to construct a single modular form to cut out all of the singularities,
(3) it was possible to find a collection of modular forms to cut out the singularities. Watson, in 1936,
[17] was the first to prove that there were a collection of modular forms cutting out the singularities of
Ramanujan’s examples. Very recently, in 2013, it was proved by Griffin, Ono, and Rolen [12], that there
was no single modular form which would cut out all of the singularities simultaneously.
Zwegers [21, 22] showed that each of Ramanujan’s examples, denoted f (z), is, up to a power of e2πiz
with z ∈ H = {x + iy : x, y ∈ R and y > 0}, the holomorphic part of some weight 1/2 harmonic Maass
forms for some modular group Γ0 (N ), denoted Mf (z), (defined by Bruinier and Funke [6], see Section 2).
∂
Additionally, ξ1/2 (Mf (z)) is a weight 3/2 cusp form where ξk := 2iy k ∂z
. This led Zagier to define a mock
modular form, called f , as the holomorphic part of any harmonic Maass form M of weight k such that
ξk (M ), called the shadow of f , is a cusp form. Thus, the mock theta functions of Ramanujan are weight
1/2 examples of Zagier’s mock modular forms. Denote the space of mock modular forms of weight k and
multiplier system χ on Γ by Mk,χ (Γ).
Our first theorem proves that any mock modular form has the three properties described by Ramanujan
in his last letter.
Theorem 1.1. Assume f (z) is a mock modular form so that Mf (z) = f (z) + f − (z) is a harmonic
Maass form of weight k ∈ 12 Z for Γ0 (N ) such that Γ0 (N )\ (H ∪ Q ∪ {∞}) has t > 1 inequivalent cusps
{q1 , . . . , qt } ⊂ Q ∪ {∞} and f − (z) is nonzero. If f − is not zero, then
(1) f (z) has exponential singularities at infinitely many rationals,
(2) for every weight k weakly holomorphic modular form g(z), f (z)−g(z) has exponential singularities
at infinitely many rationals,
Keywords: Quantum modular form, Eichler integral, mock theta function, Ramanujan, radial limit.
1
2
DOHOON CHOI, SUBONG LIM, AND ROBERT C. RHOADES
(3) there is a collection {Gj }tj=1 of weakly holomorphic modular forms such that f (z) − Gj (z) is
bounded toward all rationals equivalent to the cusp qj .
Remark 1.2. Parts (1) and (2) follow from the work of Griffin, Ono, and Rolen [12] and are based on the
results of Bruinier and Funke [6]. Part (3) is established using a generalization of a result of Borcherds
[2].
Remark 1.3. The only N for which Γ0 (N )\H has one cusp is N = 1. However, if f is a mock modular
form for Γ0 (1), then f is also a mock modular form for Γ0 (N ) for any N > 1. Thus, in fact, Theorem
1.1 implies that the Ramnujan’s claim is true for mock modular forms on Γ0 (1).
Remark 1.4. Ramanujan referred to “theta functions” rather than modular forms in his letter. However,
we understand today that the theta functions Ramanujan referred to may be taken to be weakly holomorphic
modular forms. A weakly holomorphic modular form is a meromorphic modular form whose poles (if any)
are supported at the cusps of Γ0 (N )\H.
Using Theorem 1.1, for a mock modular form f (z), associated to a harmonic Maass form on Γ0 (N )
with N > 1, define a map Qf : Q → C as the following way: if a rational number x is equivalent to qj
under the action of Γ0 (N ), then
(1.1)
Qf (x) := lim (f − Gj )(x + it).
t→0+
This map Qf is well-defined by Theorem 1.1.(3) because for each rational number x there is a unique qi
equivalent to x under the action of Γ0 (N ) (see section 3 for details). Motivated by a number of examples
from quantum invariants of 3-manifolds, Vassiliev invariants of knots, and period functions of Maass wave
forms, Zagier [20] defined a weight k quantum modular form
as a complex-valued function f on Q or
a b
possibly P1 (Q) \ S for some subset S, such that for all γ =
∈ SL2 (Z) the function
c d
az + b
−k
,
hγ,χ (x) := f (x) − (f |k,χ γ) (x), where (f |k,χ γ)(z) := χ(γ)(cz + d) f
cz + d
satisfies some property of continuity or analyticity (see section 2 for more details). Let Qk,χ (Γ) be the
space of quantum modular forms f (x) such that for every γ ∈ Γ the function defined by f (x) − (f |k,χ γ)(x)
can be extended smoothly to R except finitely many points. The following theorem shows that we have
a linear map from the space of mock modular forms on Γ0 (N ) to the space of quantum modular forms
! (Γ (N )).
whose kernel is the space of weakly holomorphic modular forms, denoted Mk,χ
0
Theorem 1.5. Let k ∈ 21 Z. Let Γ = Γ0 (N ) with N > 1 and χ be a multiplier system of weight k on Γ.
Then we have a linear injective map
!
µ : Mk,χ (Γ)/Mk,χ
(Γ) → Qk,χ (Γ)
given by µ(f ) = Qf .
Remark 1.6. Hints of the quantum modular properties of the mock theta functions appear in Ramanujan’s letter. The quantum modular properties of the mock theta functions in the context of Ramanujan’s
examples is discussed by Lawerence and Zagier [15] based on observations of Zwegers (see also [20]) and
is described by Folsom, Ono, and the third author [10].
Remark 1.7. A second approach an injective map is to proceed by identifying a mock modular form with
its shadow. The Eichler integral of the shadow extends to a subset of Q and produces a quantum modular
form. This perspective, in the case of mock modular forms with shadows given by a unary theta function,
is considered in the recent work of Bringmann, Creutzig, and Rolen [3].
MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS
3
One of the first examples of a quantum modular form is the strange function of Kontsevich [18] defined
by
F (q) = 1 + (1 − q) + (1 − q)(1 − q 2 ) + (1 − q)(1 − q 2 )(1 − q 3 ) + · · ·
which is defined only at roots of unity. Zagier computed and proved
e
−u/24
−u
F (e
∞
X
23 u 1681 u 2 257543 u 3
Tn u n
) =1 +
+
+ ··· =
+
1! 24
2!
24
3!
24
n! 24
n=0
where Tn , defined by Glaisher in 1898, are the “algebraic” part of certain L-values
12
(2n + 1)! 6 2n+2
√
·L
, 2n + 2
Tn =
π
·
2 3
where 12
n is 1 if n ≡ ±1 (mod 12), −1 if n ≡ ±5 (mod 12), and 0 otherwise. The following result shows
that the behavior of a period function of the nonholomorphic Eichler integral associated with a shadow
g(z) of a mock modular form f (z) has similar behavior. We let
Z i∞
k−1
g(τ )(τ − z̄)−k dτ
g̃(z) := (−2i)
z
for z ∈ H. Then Qf (x) is the same as the limit point g̃(z) at z = x for x ∈ Q.
Theorem 1.8. Let k ∈ 21 Z, Γ = Γ0 (N ) with N > 1 and χ be a multiplier system of weight k on Γ. For
γ = ac db ∈ Γ with c > 0 and a rational number x0 > − dc
∞
X
(k)l
d
k−1
g̃(x0 + it) − (g̃|k,χ γ)(x0 + it) = (−2i)
Φ g, − , x0 ; −k − l (it)l ,
l!
c
l=0
R∞
where Φ(g, a, b; s) = a g(τ )(τ − b)s dτ and (a)n is the Pochhammer symbol defined by (a)n = a(a +
1) . . . (a + n − 1) for n ≥ 1 and (a)0 = 1.
Remark 1.9. The above theorem gives the same formulas for mock modular forms since
f (z) − (f |k,χ γ)(z) = g̃(z) − (g̃|k,χ γ)(z).
Zagier’s computations were motivated by computations of Kontsevich [18]
3
2πi
3
2πi 2 19
2πi 3 207
2πi 4
−2πik
2πi/k
F (e
) ∼ ζ(k)k 2 F (e
)+1+ −
+
−
+
−
+
−
+ ···
k
2!
k
3!
k
4!
k
where ζ(k) is a root of unity that depends on k.
Remark 1.10. The coefficients of this expansion agree with those computed by Zagier for F (e−t ).
The following results generalizethis for quantum modular
forms associated to mock modular forms.
P
d
The twisted L-function L g, − c ; s associated to g(z) = m+κ>0 a(m)e2πi(m+κ)z is given by the analytic
continuation of
X a(m)e2πi(m+κ)(− dc )
m+κ>0
(see Lemma 1 in [13]).
(m + κ)s
4
DOHOON CHOI, SUBONG LIM, AND ROBERT C. RHOADES
Theorem 1.11. Let k ∈ 12 Z, Γ = Γ0 (N ) with N > 1 and χ be a multiplier system of weight k on Γ.
Suppose that f (z) is a mock modular form of weight k and multiplier system χ on Γ whose shadow is
m
g(z) ∈ S2−k,χ̄ (Γ). Let x0 = −d
c ∈ Q such that (c, d) = 1, N |c and c > 0. Assume that χ(T ) = 1 for a
positive integer m. Then
k
∞
a −1 1−k 1 k X
1
1
(k)l Γ(1 − k − l)
∼
χ̄(γ)Q
Qf x 0 +
+
f
2
mnc
mnc
c
4π
mnc
l!
l=0
l
−2πi
×L (g, x0 ; 1 − k − l)
mnc2
as n → ∞, where γ = ac db ∈ Γ.
1.1. An example. In his letter, Ramanujan defined
f (q) = 1 +
q
q4
+
+ ··· .
(1 + q)2 (1 + q)2 (1 + q 2 )2
It is clear that f (q) has singularities exactly when q is an even order root of unity. Ramanujan compared
f (q) to b(q) = (1−q)(1−q 3 )(1−q 5 ) · · · (1−2q+2q 4 −· · · ). The function q −1/24 b(q) is a weakly holomorphic
modular form. Moreover, Ramanujan claimed, and Watson [17] proved, that f (q)−b(q) = O(1) as q heads
radially toward a fourth root of unity and f (q) − (−b(q)) = O(1) has q heads radially toward a root of
unity that is 2 modulo 4. Thus the pair b(q), −b(q) cuts out the singularities of f (q). Zwegers [21] showed
that with q = e2πiz , q −1/24 f (q) is the holomorphic part of a weight 1/2 harmonic Maass form. More
precisely, Bringmann and Ono [4] (see Corollary 2.3) established that q −1 f (q24 ) is the holomorphic part
of a weight 1/2 weak Maass form on Γ0 (144) with Nebentypus character 12· .
With q = e2πix

−1/24 f (q)

q is an odd order root of unity,
q
Qf (x) = q −1/24 (f (q) − b(q)) q is a 4kth root of unity,

 −1/24
q
(f (q) + b(q)) q is a (4k+2)th root of unity,
where the right hand side is interpreted as the radial limit as q approaches the root of unity. The following
gives a small table of values for Qf (ζn ) where ζn := e2πi/n .
n
ζ24n Qf (ζn )
1 2
4
3 4
3
4
3 (1
−
ζ32 )
4
4i
5
4 3
3 (ζ5
−
ζ54 )
6
−4ζ6
Moreover, we have the asymptotic expansions
!
2
3
n
∞
1
5
153
t
12285
t
4 X αn
t
t
Qf
=4
+
+ 3
−
+ 4
−
+ ··· = √
−
−
3 32
24
3 · 2!
24
3 · 3!
24
6
3 n=0 n!
!
2
3
n
∞
X βn
1
t
t
3985
t
1743623
t
t
Qf
−
=4 1 + 23 −
+
−
+
−
+ ··· = 4
−
2 2πi
24
2!
24
3!
24
n!
24
n=0
t
−
2πi
R∞
cosh( πx
3 )
where αn = 0 x2n cosh(πx)
dx and βn are given in Theorem 4.1 of [11] in terms of the Euler numbers.
Zwegers [22] showed that the integrals appearing in the definition of αn and βn may be written in terms
of integrals of weight 3/2 unary theta functions.
1.2. Roadmap. Section 2 contains background on modular forms, harmonic Maass forms and quantum
modular forms. Section 3 contains the proofs of the main theorems.
MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS
5
Acknowledgements
The present work started during a visit of the third author to the Korean Institute for Advanced Study.
The authors thank the institute for its hospitality.
2. Modular objects
In this section we recall definitions and basic facts about modular forms and harmonic Maass forms.
Let k ∈ 21 Z and let Γ = Γ0 (N ) with N > 1. Let χ be a multiplier system of weight k on Γ, i.e., χ : Γ → C
satisfies the following conditions:
(1) |χ(γ)| = 1 for all γ ∈ Γ.
(2) For z ∈ H, χ satisfies the consistency condition
χ(γ3 )(c3 z + d3 )k = χ(γ1 )χ(γ2 )(c1 γ2 z + d1 )k (c2 z + d2 )k ,
where γ3 = γ1 γ2 and γi = acii dbii for i = 1, 2, 3.
We recall the slash operator
(f |k,χ γ)(z) := χ̄(γ)(cz + d)−k f (γz)
for any function f (z) on H and γ = ac db ∈ Γ. Note that T := ( 10 11 ) generate the subgroup Γ∞ of
translations in Γ. If a holomorphic function f (z) satisfies (f |k,χ T )(z) = f (z), then f (z) has the Fourier
expansion at i∞:
∞
X
(2.1)
f (z) =
a(n)e2πi(n+κ)z ,
n=−∞
where κ is a constant in [0, 1) such that χ(T ) = e2πiκ . Suppose that Γ has t ≥ 1 inequivalent cusps.
Let q1 = i∞ and q2 , . . . , qt be the inequivalent parabolic cusps on the boundary of H. Suppose also that
∗ ∗
Qj = cj dj is a generator of Γj , which is the cyclic subgroup of Γ fixing qj for 2 ≤ j ≤ t. For 2 ≤ j ≤ t,
put χ(Qj ) = e2πiκj , 0 ≤ κj < 1. If a holomorphic function f (z) satisfies (f |k,χ Qj )(z) = f (z), then f (z)
has the Fourier expansion at qj :
(−z)−k f (A−1
j z) =
(2.2)
∞
X
aj (n)e2πi(n+κj )z/λj ,
n=−∞
−1
where Aj = 01 −q
and λj is a positive real number called the width of the cusp qj , which is chosen so
j
1 λj
that A−1
Aj generates Γj . For convenience, let A1 = ( 10 01 ).
j
0 1
Definition 2.1. Suppose f (z) is holomorphic in the upper half plane H and satisfies the functional
equation (f |k,χ γ)(z) = f (z) for all γ ∈ Γ.
(1) If f (z) has only finitely many terms with n + κ < 0 in (2.1) and with n + κj < 0, 2 ≤ j ≤ t, in
(2.2), then f (z) is called a weakly holomorphic modular form of weight k and multiplier system χ
! (Γ).
on Γ. The set of all such weakly holomorphic modular forms is denoted by Mk,χ
!
(2) Let f (z) ∈ Mk,χ (Γ). Suppose in addition f (z) has only terms with n + κ ≥ 0 in (2.1) and
n + κj ≥ 0, 2 ≤ j ≤ t, in (2.2). Then f (z) is called a holomorphic modular form. The set of
! (Γ) is denoted by M
holomorphic modular forms in Mk,χ
k,χ (Γ).
(3) If f (z) ∈ Mk,χ (Γ) and has only terms with n + κ > 0, n + κj > 0 in the expansions (2.1), (2.2),
respectively, then f (z) is called a cusp form. The collection of cusp forms in Mk,χ (Γ) is denoted
by Sk,χ (Γ).
6
DOHOON CHOI, SUBONG LIM, AND ROBERT C. RHOADES
A harmonic Maass form of weight k and multiplier system χ on Γ is a smooth function on H with
possible singularities at cusps that transforms like a modular form and is annihilated by the weight k
hyperbolic Laplacian
2
∂
∂
∂2
∂
2
+ iky
.
∆k := −y
+
+i
∂x2 ∂y 2
∂x
∂y
Now we give the precise definition of a harmonic Maass form.
Definition 2.2. A harmonic Maass form of weight k and multiplier system χ on Γ is any smooth function
on H satisfying
(1) (f |k,χ γ)(z) = f (z) for all γ ∈ Γ,
(2) ∆k f = 0,
(3) a linear exponential growth condition in terms of y at every cusp.
We write Hk,χ (Γ) for the space of harmonic Maass forms of weight k and character χ on Γ.
Recall that any harmonic Maass form f (z) of weight k has the unique decomposition f (z) = f + (z) +
where
X
f + (z) =
(2.3)
a+ (n)e2πi(n+κ)z/λ ,
f − (z),
n−∞
−
f (z) = δκ,0 a− (0)y 1−k +
X
a− (n)Γ(4π(n + κ)y/λ, −k + 1)e2πi(n+κ)z/λ .
n∞
n+κ6=0
Here, δκ,0 = 1 if κ = 0, and δκ,0 = 0 otherwise (for this decomposition see [6, section 3]). The function
f + (z) (resp. f − (z)) is called the holomorphic (resp. nonholomorphic) part of f (z). By Zagier’s definition
in [19], f + (z) is called a mock modular form and f (z) is its completion.
There is a differential operator ξk := 2iy k ∂∂z̄ which plays an important role in the theory of harmonic
Maass forms. The assignment f (z) 7→ ξk (f )(z) defines an anti-linear mapping
!
ξk : Hk,χ (Γ) → M2−k,
χ̄ (Γ).
! (Γ) (see [6, Proposition 3.2]). We let H ∗ (Γ) denote the inverse image
Moreover, the kernel of ξk is Mk,χ
k,χ
∗ (Γ), then the Fourier
of the space of cusp forms S2−k,χ̄ (Γ) under the mapping ξk . Hence, if f (z) ∈ Hk,χ
−
∗ (Γ) the
coefficients a (n) vanish if n+κ is nonnegative. It is known for a harmonic Maass form f (z) ∈ Hk,χ
nonholomorphic part f − (z) is given by the nonholomorphic Eichler integral associated with the shadow
g(z) := ξk (f ) (see [5] or [8])
Z i∞
−
k−1
(2.4)
f (z) = −(−2i)
g(τ )(τ − z̄)−k dτ .
z
z̄)−k
Here, we determine the branch of (τ −
by means of the convention z −k = |z|−k e−ik arg z , where
−π ≤ arg z < π.
A quantum modular form is a new modular object introduced by Zagier [20]. He showed that a quantum
modular form gives a deep understanding for various objects such as quantum field theory and invariants
of knots. Following Zagier’s notion of a quantum modular form, we define a quantum modular form.
Definition 2.3. A quantum modular form of weight k and multiplier system χ on Γ is a function f (x)
on Q such that for every γ ∈ Γ the function defined by
pγ (x) := f (x) − (f |k,χ γ)(x)
MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS
7
extends to a function on R except finitely many points and has some property of continuity or analyticity.
We call pγ (x) a period function of f (x).
3. Proof of the main theorems
In this section we prove the main results. To prove Theorem 1.1 we need the following lemma.
Lemma 3.1. Let k ∈ 12 Z and Γ = Γ0 (N ). We take a cusp qj in {q1 , . . . , qt }. Suppose that f (z) is a
∗ (Γ) which has a Fourier expansion at a cusp q of the form
harmonic Maass form in Hk,χ
j
X
X
2πi(n+κj )z/λj
2πi(n+κj )z/λj
a−
.
a+
+
q (n)Γ(−4π(n + κj )y/λj , −k + 1)e
q (n)e
n+κj <0
n+κj >0
Let Cj be a set of rationals which are equivalent to the cusp qj under the action of Γ. Then for any x ∈ Cj
Z i∞
+
−
k−1
(3.1)
lim f (x + it) = − lim f (x + it) = (−2i)
g(τ )(τ − x)−k dτ ,
t→0+
t→0+
x
where g(z) := ξk (f ).
Proof. Let x = γqj for some γ =
a b
c d
in Γ0 (N ). Since g(z) is a cusp form, we have by (2.4)
Z i∞
−
k−1
− lim f (x + it) = lim g̃(x + it) = (−2i)
g(τ )(τ − x)−k dτ .
t→0+
t→0+
x
Thus the proof can be completed by showing limt→0+ f (x + it) = 0. To do it we consider the action of
Aj γ −1 for a special geodesic:
(3.2)
Aj γ −1 (x + it) =
Aj γ −1 (x + it) =
1
(c(cqj
(cqj +d)2
1
(−cd + i 1t )
c2
+ d) + i 1t )
if j 6= 1,
if j = 1.
−1
Recall that Aj = 01 −q
if j 6= 1 and that Aj = ( 10 01 ) if j = 1. In the first equation cqj + d is not zero
j
since qj is not equivalent to i∞ under the action of Γ0 (N ), and in the second equation c is also not zero
since x ∈ Q and x = γi∞. Moreover, a modular property of f implies
−1
f (x + it) = f |k,χ γ −1 (x + it) = χ̄(γ −1 )(−c(x + it) + d)−k f (A−1
j Aj γ (x + it))
Combining this with (3.2), we obtain limt→0+ f (x + it) = 0 since by the assumption f (A−1
j z) decays
+
exponentially when z approaches to i∞. Since limt→0+ f (x+it) = limt→0+ f (x+it)−limt→0+ f − (x+it),
we complete the proof.
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. Parts (1) and (2) follow from the work of Griffin, Ono, and Rolen [12] and are
based on the results of Bruinier and Funke [6]. Now we prove part (3) using Lemma 3.1. Suppose that f (z)
! (Γ)
is a mock modular form in Mk,χ (Γ). We can find a weakly holomorphic modular form Gj (z) ∈ Mk,χ
such that (f − Gj )(z) has neither a principal part nor a constant term in the Fourier expansion at a cusp
qj by a variant of Theorem 3.1 in [2]. Then by Lemma 3.1 a function (f − Gj )(z) is bounded toward all
rationals equivalent to the cusp qj .
Suppose that f (z) is a mock modular form whose shadow is g(z) ∈ S2−k,χ̄ (Γ0 (N )) and that x is a
rational. Then x is equivalent to a cusp qj for some 1 ≤ j ≤ t and by Theorem 1.1 (3) there is a weakly
! (Γ (N )) which has the same singularity and constant term with
holomorphic modular form Gj (z) in Mk,χ
0
a mock modular form f (z) at the cusp qj . Although Gj (z) is not unique, Lemma 3.1 shows that the value
8
DOHOON CHOI, SUBONG LIM, AND ROBERT C. RHOADES
limx→0+ (f − Gj )(x + it) for x ∈ Cj is the same as g̃(x) and hence it is independent of the choice of Gj (z).
Therefore, we have a well-defined function Qf on Q which is given by Qf (x) = limx→0+ (f − Gj )(x + it)
for x ∈ Q.
The next theorem shows that Qf is a quantum modular form for any mock modular forms f .
Theorem 3.2. Let k ∈ 21 Z. Let Γ = Γ0 (N ) with N > 1 and χ be a multiplier system of weight
k on Γ.
a
b
If f (z) is a mock modular form of weight k and multiplier system χ on Γ, then for γ = c d ∈ Γ the
function Qf (x) satisfies
(Z
)
Z
i∞
Qf (x) − (Qf |k,χ γ)(x) = (−2i)k−1
x
g(τ )(τ − x)−k dτ − δk,γ,x
x
g(τ )(τ − x)−k dτ ,
γ −1 (i∞)
where g(z) is the shadow of f (z) and
δk,γ,x
(
e2πik
=
1
if c ≤ 0 and cx + d < 0,
otherwise.
Moreover, if c > 0 or cx + d > 0 or k is an integer, then
Z
Qf (x) − (Qf |k,χ γ)(x) = (−2i)k−1
i∞
g(τ )(τ − x)−k dτ .
γ −1 (i∞)
Hence, Qf (x) is a quantum modular form of weight k and multiplier system χ on Γ.
To prove the theorem we need the following lemma on analytic property for period functions of Qf .
Lemma 3.3. Under the same notations as in Theorem 3.2 let
or a half integer and
R i∞m denote an integer
m
pγ,m (x) be a C-valued function on R defined by pγ,m (x) := γ −1 (i∞) g(τ )(τ − x) dτ . Then pγ,m (x) is a
smooth function on R and has an analytic extension to {u + iv ∈ C | u > 0 or v ≤ 0}.
The proof of this lemma can be easily obtained by standard methods and direct computations. Thus
we give just a sketch of the proof.
Proof of Lemma 3.3. To prove smoothness of pγ,m (x) it is enough to show that the derivative of pγ,m (x)
is mpγ,m−1 (x) if m 6= 0. Let x0 := γ −1 (i∞). Taking a special contour, we have
Z ∞
(pγ,m (x + t) − pγ,m (x))
(x0 + ir − z0 − t)m − (x0 + ir − z0 )m
lim
= lim
g(x0 + ir)
idr.
t→0
t→0 0
t
t
Assume that t is sufficiently small. By direct computation we have
((γ −1 (i∞) + ir − x − t)m − (γ −1 (i∞) + ir − x)m )t−1
has polynomial growth independent of t when r approaches to ∞ or 0. Since g(z) is a cusp form,
g(γ −1 (i∞) + ir)((γ −1 (i∞) + ir − x − t)m − (γ −1 (i∞) + ir − x)m )t−1
decays exponentially independent of t as r goes to 0 or ∞. Thus by lebesgue’s dominated convergence
theorem we can exchange the order of limit and integration. This implies smoothness of pγ,m (x). Considering the branch of (τ − z̄)m , one can prove by a similar argument to the above that pγ,m (x) has an
analytic extension to {u + iv ∈ C | u > 0 or v ≤ 0}. Thus we omit it.
Now we prove Theorem 3.2.
MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS
9
R i∞
Proof of Theorem 3.2. From Lemma 3.1 we know that Qf (x) = (−2i)k−1 x g(τ )(τ − x)−k dτ . Thus
by the definition of the slash operator we have
Z i∞
k−1
−k
g(τ )(τ − γx)−k dτ .
(Qf |k,χ γ)(x) = −(−2i) χ(γ)(cx + d)
γx
b
So we can compute the function (Qf |k,χ γ)(x) for γ = ac d ∈ Γ using the argument in Lemma 2.2 of [14].
Using the substitution τ → γτ in the above integral and the fact that g(z) is a cusp form in S2−k,χ̄ (Γ),
we see that
Z x
(Qf |k,χ γ)(x) = (−2i)k−1
(cx + d)−k (cτ + d)−k g(τ )(γτ − γx)−k dτ .
γ −1 (i∞)
By the simple computation we have
τ −x
(cτ +d)(cx+d)
τ −x
(cτ + d)(cx + d)
= γτ − γx. Since τ − x and γτ − γx are in H, we see that
−k
=
(τ − x)−k
.
{(cτ + d)(cx + d)}−k
−1
(cτ + d)−k (cx + d)−k , where
Furthermore, by comparing arguments we have {(cτ + d)(cx + d)}−k = δk,γ,x
(
e2πik if c ≤ 0 and cx + d < 0,
δk,γ,x =
1
otherwise.
Rx
From this we see that (Qf |k,χ γ)(x) = (−2i)k−1 δk,γ,x γ −1 (i∞) g(τ )(τ − x)−k dτ . Thus if c > 0 or cx + d > 0
or k is an integer, then
Z i∞
k−1
g(τ )(τ − x)−k dτ .
Qf (x) − (Qf |k,χ γ)(x) = −(−2i)
γ −1 (i∞)
Finally, to complete the proof, it must be shown that period functions of Qf are smooth on R except
finitely many points. Lemma 3.3 implies Qf (x) − (Qf |k,χ γ)(x) is smooth on R if c > 0. Let γ1 := N1 01
0
and γ2 := −1
N −1 . Then −I = γ1 γ2 . Thus Qf (x) − (Qf |k,χ − γ)(x) can be expressed as the following
form
Qf (x) − (Qf |k,χ γ2 )(x) + (Qf − Qf |k,χ γ1 )|k,χ γ2 (x) + (Qf − Qf |k,χ γ)|k,χ γ1 γ2 (x).
Therefore, Qf (x) − (Qf |k,χ γ)(x) is smooth on R except finitely many points.
Thanks to the Eichler-Shimura coholomogy, we can prove by using Theorem 3.2 that our map µ :
! (Γ) → Q
Mk,χ (Γ)/Mk,χ
k,χ (Γ) is injective.
Proof of Theorem 1.5. It is shown that Qf is well-defined for mock modular forms f ∈ Mk,χ (Γ).
! (Γ). Thus the map
Moreover, from the definition of Qf it is clear that Qf is a zero function if f ∈ Mk,χ
! (Γ) → Q
µ : Mk,χ (Γ)/Mk,χ
k,χ (Γ) is well-defined.
Let g(z) be the shadow of f . The nonholomorphic part of Mf is given by a nonholomorphic Eichler
integral associated with g(z) as in (2.4). Thus for the injectivity of µ it is sufficient to show that if Qf is
a zero function, then g(z) is also a zero function. Suppose that Qf (x) is a zero function. Let γ0 := ac db
be an element in Γ with c > 0. Then by Theorem 3.2 we have
Z i∞
k−1
Qf (x) − (Qf |k,χ γ0 )(x) = −(−2i)
g(τ )(τ − x)−k dτ .
γ0−1 (i∞)
10
DOHOON CHOI, SUBONG LIM, AND ROBERT C. RHOADES
R i∞
Since Qf (x) = 0 on Q by the assumption, a function γ −1 (i∞) g(τ )(τ − z̄)−k dτ is zero on the real line. On
0
the other hand, Lemma 3.3 implies this function is holomorphic on E := {u + iv ∈ C | u > 0 or v ≥ 0}.
Thus this function should be a zero function on H, H ⊂ E, by the identity theorem for holomorphic
functions.
From the Eichler-Shimura cohomology theory there is an isomorphism η between S2−k,χ̄ (Γ) and the
1
Eichler cohomology H̃−k,χ
(Γ). The image of g nuder the map η is a 1-cocycle element defined by <
R i∞
gγ (z)| γ ∈ Γ >, where gγ (z) is a function on H defined by γ −1 (i∞) g(τ )(τ − z̄)−k dτ and < gγ (z)|γ ∈ Γ >
is the class determined by the cocycle {gγ (z)| γ ∈ Γ} (for more details, see [14, Section 2]). We already
checked that gγ (z) is a zero function if γ = ac db ∈ Γ with c > 0 or γ = T . Moreover, since −I =
−1 0 1 0
a b
N 1
N −1 , we see Γ can be generated by T and some elements γ = c d ∈ Γ with c > 0. Thus gγ (z)
is a zero function for all γ ∈ Γ by the cocycle relation. Since η is an isomorphism, this implies that g(z)
is a zero function. It is clear that µ is linear and so we complete the proof.
Using analytic property for period functions of Qf , we prove Theorem 1.8 and 1.11.
Proof of Theorem 1.8. By Theorem 3.2 we have
k−1
Z
g̃(z) − (g̃|k,χ γ)(z) = (−2i)
i∞
g(τ )(τ − z̄)−k dτ
γ −1 (i∞)
for z ∈ H. Thus, since x0 6= γ −1 i∞, it is analytic at z = x0 . So if we let pγ (z) = g̃(z) − (g̃|k,χ γ)(z), then
pγ (z) =
∞
(l)
X
pγ (x0 )
l=0
l!
(z − x0 )
(l)
around x0 . Here, pγ (x0 ) denotes the lth derivative of pγ (x) at x = x0 . Since
Z i∞
(l)
k−1
pγ (x0 ) = (−2i) (−k)l
g(τ )(τ − x0 )−k−l dτ ,
− dc
we have
k−1
g̃(x0 + it) − (g̃|k,χ γ)(x0 + it) = (−2i)
Z
∞
X
(−k)l
l=0
l!
i∞
− dc
This completes the proof.
g(τ )(τ − x0 )−k−l dτ (it)l .
1
Proof of Theorem 1.11. Let let pγ (x) := Qf (x) − (Qf |k,χ γ)(x) and zn := x0 + mnc
2 for integers n.
1
−1
−mn
Note that zn → x0 as n → ∞ and zn = (γ T
γ)(i∞). Therefore, we have czn + d = mnc
and
a
Qf (γzn ) = Qf ((T −mn γ)(i∞)) = Qf (γ(i∞)) = Qf
c
m
because Qf (T x) = Qf (x) for any x ∈ Q and m ∈ Z. Therefore, we obtain
−k
a
1
1
(3.3)
pγ (zn ) = Qf x0 +
Q
.
−
χ̄(γ)
f
mnc2
mnc
c
On the other hand, by Theorem 3.2 and Lemma 3.3 pγ (x) is smooth at x = x0 . Hence by Taylor’s
Theorem we have an asymptotic expansion at x = x0
pγ (x) ∼
∞
(l)
X
pγ (x0 )
l=0
l!
(x − x0 )l
MOCK MODULAR FORMS AND QUANTUM MODULAR FORMS
11
(l)
as x → x0 . Here, pγ (x0 ) denotes the lth derivative of pγ (x) at x = x0 .
If s is a complex number whose the real part is sufficiently large, then
Z ∞
Z i∞
Z i∞
s
1+s
s
g(iy + x0 )y s dy
g(τ + x0 )τ dτ = i
g(τ )(τ − x0 ) dτ =
0
0
x0
1+s
= i
L (g, x0 ; 1 + s) Γ(1 + s)
1
2π
1+s
.
A function L (g, x0 ; 1 + s) Γ(1 + s) has an analytic continuation to C (see Lemma in [13]). Using the
integral expression of pγ (x) in Theorem 3.2, we have
Z i∞
k−1
p(l)
(x
)
=
(−2i)
(−k)
g(τ )(τ − x0 )−k−l dτ
0
l
γ
x0
1−k−l
=i
L (g, x0 ; 1 − k − l) Γ(1 − k − l)
1
2π
1−k−l
.
Therefore,
(3.4)
pγ (zn ) ∼
−1
4π
1−k X
∞
l=0
(−k)l Γ(1 − k − l)
L(g, x0 ; 1 − k − l)
l!
Combining (3.3) and (3.4), we get the desired result.
−2πi
mnc2
l
.
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School of liberal arts and sciences, Korea Aerospace University, 200-1, Hwajeon-dong, Goyang, Gyeonggi
412-791, Republic of Korea,
School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Seoul 130-722, Republic of
Korea
E-mail address: [email protected]
School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Seoul 130-722, Republic of
Korea
E-mail address: [email protected]
Center for Communications Research, 805 Bunn Dr., Princeton, NJ 08450
E-mail address: [email protected]