Cohomological relation between Jacobi forms and skew-holomorphic Jacobi forms

0.22 MB·English

by  Dohoon Choi

COHOMOLOGICAL RELATION BETWEEN JACOBI FORMS AND
SKEW-HOLOMORPHIC JACOBI FORMS
DOHOON CHOI AND SUBONG LIM
3
1 Abstract. Eichler and Zagier developed a theory of Jacobi forms to understand and extend
0 Maass’ work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic
2 Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi

n forms. Inthispaper,weexplainarelationbetweenholomorphicJacobiformsandskew-holomorphic
a Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from
J the direct sum of the space of Jacobi cusp forms on ΓJ and the space of skew-holomorphic Jacobi

6 cusp forms on ΓJ with the same half-integral weight to the Eichler cohomology group of ΓJ with
1
a coefficient module coming from polynomials.

]
T
N
. 1. Introduction
h
t
a
Let H be the complex upper half plane. A Jacobi form is a function of two variables τ
m
∈
H and z C, which satisfies modular transformation properties with respect to τ and elliptic
[ ∈
transformation properties with respect to z. Eichler and Zagier [10] systematically developed

1
a theory of Jacobi forms. The theory of Jacobi forms has grown enormously since then with
v
4 beautiful applications in many areas of mathematics and physics, for example, the Saito-Kurokawa
9 conjecture (see [26]), mock theta functions (see [28]), the theory of Donaldson invariants of CP2
5
that are related to gauge theory (see [12]) and the Mathieu moonshine (see [8]). On the other
3
. hand, skew-holomorphic Jacobi forms are introduced by Skoruppa [24, 25]. They play a crucial
1
0 role in understanding liftings of modular forms and Jacobi forms. Moreover, a Jacobi form and
3
a skew-holomorphic Jacobi form have deep connections to modular forms through the beautiful
1
: theory of the theta expansion, developed by Eichler and Zagier [10], which gives an isomorphism
v
between (skew-holomorphic) Jacobi forms and vector-valued modular forms.
i
X
Our aim is to explain the relation between holomorphic Jacobi forms and skew-holomorphic
r
a Jacobi forms in terms of the Eichler cohomology group of a Jacobi group. Eichler in [9] conceived
the Eichler cohomology theory while studying generalized abelian integrals, which are now called
the Eichler integrals, and proved that the direct sum of two spaces of cusp forms on Γ with the
same integral weight is isomorphic with the first cohomology group of Γ with a certain module
of polynomials as a coefficient if Γ is a finitely generated Fuchsian group of the first kind which
has at least one parabolic class. After the results of Eichler, the Eichler cohomology theory was
studied further by Gunning [13], Husseini and Knopp [14] and Lehner [23].
Inthispaper, wedealwiththeEichlercohomologytheoryforJacobiformsandskew-holomorphic
Jacobi forms with a half-integral weight using a coefficient module analogous to the module of
polynomials used by Eichler in [9]. More precisely, let Γ SL(2,Z) be a H-group. Here, an
⊂
Keynote: Jacobi form, Eichler cohomology.
2010 Mathematics Subject Classification: 11F50, 11F67 .
1
2 DOHOONCHOI ANDSUBONGLIM
H-group is a finitely generated Fuchsian group of the first kind, which has at least one parabolic
class. We assume that I Γ and [SL(2,Z) : Γ] < . Let k Z + 1 and χ be a (unitary)
− ∈ ∞ ∈ 2
multiplier system of weight k on Γ. We denote the vector space of Jacobi cusp forms (resp. skew-
holomorphic Jacobi cusp forms) of weight k, index m and multiplier system χ on ΓJ by S (ΓJ)
k,m,χ
(resp. Jsk,cusp(ΓJ)), where ΓJ denotes the Jacobi group Γ⋉Z2.
k,m,χ
Now we consider the following special C[ΓJ]-module to define a cohomology group of ΓJ.
Definition 1.1. Let Pe be the set of holomorphic functions g(τ,z) on H C which satisfy the
k,m ×
following conditions
(1) for all X Z2, (g X)(τ,z) = g(τ,z),
m
∈ |
(2) Lk+1g = 0.
m
This set Pe is preserved under the slash operator (γ,X) for a non-zero integer k
k,m |−k+21,m,χ
and (γ,X) ΓJ and forms a vector space over C (see section 3.1 for the detail of slash opera-
∈
tor). Using the space Pe as a coefficient module, we can define a parabolic cohomology group
k,m
H˜1 (ΓJ,Pe ) (for the precise definition see section 3.3). In the following theorem we give a
−k+1,m,χ k,m
2
group cohomological relation between Jacobi cusp forms and skew-holomorphic Jacobi cusp forms
of half-integral weights. In other words, we prove the existence of an isomorphism between the
parabolic cohomology group of ΓJ and the direct sum of the space of Jacobi cusp forms on ΓJ and
the space of skew-holomorphic Jacobi cusp forms on ΓJ.
Theorem 1.2. For a positive integer k, index m Z with m > 0 and multiplier system χ of
∈
weight k + 1, we have an isomorphism
− 2
η˜: S (ΓJ) Jsk,cusp (ΓJ) = H˜1 (ΓJ,Pe ).
k+2+21,m,χ ⊕ k+2+21,m,χ ∼ −k+21,m,χ k,m
One can be interested in finding a cohomology group isomorphic to the space of cusp forms
itself. In this direction it was studied in [13, 20]. Especially Knopp [16] established anisomorphism
between the space of cusp forms with a real weight on Γ and the Eichler cohomology group with
a special coefficient module. We would like to point out that in these cases the coefficient module
is the space of holomorphic functions satisfying a certain growth condition on which Γ acts in a
manner analogous to the action of Γ on vector spaces of polynomials.
For Jacobi forms Choie and the second author in [7] introduced the Jacobi integrals, which are
analogues of Eichler integrals. Authors of [7] also investigated their properties and gave various ex-
amples of Jacobi integrals including the construction of Jacobi integrals using the theta expansion.
In [4] authors studied the Eichler cohomology theory associated with Jacobi forms of arbitrary
real weight using a coefficient module e of holomorphic functions on H C satisfying a certain
Pm ×
growth condition and proved the existence of an isomorphism between the cohomology group of
ΓJ and the space of Jacobi cusp forms on ΓJ with a real weight.
The remainder of this paper is organized as follows. In section 2, we introduce vector-valued
modular forms, review the supplementary function theory in terms of vector-valued Poincar´e serie
and describe how to construct the vector-valued Eichler integral for given parabolic cocycle. In
section 3, we give basic notions of Jacobi forms and skew-holomorphic Jacobi forms and we define
the cohomology group of ΓJ with a coefficient module Pe . In section 4, we prove Theorem 1.2.
k,m
COHOMOLOGICAL RELATION BETWEEN JACOBI FORMS AND SKEW-HOLOMORPHIC JACOBI FORMS 3
2. Vector-valued Modular Forms
In this section, we introduce the basic notions of vector-valued modular forms and vector-valued
Poincar´e series. With this vector-valued Poincar´e series we review the supplementary function
theory which was established by Knopp and his collaborators in [14, 15]. We also describe how to
construct the vector-valued Eichler integral for a given parabolic cocycle.
2.1. Vector-valued modular forms. Webeginbyintroducing thedefinitionofthevector-valued
modular forms. Let k Z and χ be a character of Γ. Let p be a positive integer and ρ : Γ
∈ →
GL(p,C) a p-dimensional unitary complex representation. We denote the standard basis elements
of the vector space Cp by e for 1 j p. With these setups, the definition of the vector-valued
j
≤ ≤
modular forms are given as follows.
Definition 2.1. A vector-valued weakly holomorphic modular form of weight k, multiplier system
χ and type ρ on Γ is a sum f(τ) = p f (τ)e of functions holomorphic in H satisfying the
j=1 j j
following conditions:
P
(1) for all γ = (a b) Γ, we have (f γ)(τ) = f(τ),
c d ∈ |k,χ,ρ
(2) for each γ = (a b) SL(2,Z), the function (cτ +d)−kf(γτ) has the Fourier expansion of
c d ∈
the form
p
(cτ +d)−kf(γτ) = a (n)e2πi(n+κj,γ)τ/λγe ,
j,γ j
j=1 n≫−∞
X X
where κ (resp. λ ) is a constant which depends on j and γ (resp. γ).
j,γ γ
Here, the slash operator γ is defined by
k,χ,ρ
|
(f γ)(τ) = χ(γ)−1(cτ +d)−kρ−1(γ)f(γτ),
k,χ,ρ
|
for γ = (a b) Γ, where γτ = aτ+b. The space of all vector-valued weakly holomorphic modular
c d ∈ cτ+d
forms f(τ) of weight k, multiplier system χ and type ρ on Γ is denoted by M! (Γ). There are
k,χ,ρ
subspaces M (Γ) and S (Γ) of vector-valued holomorphic modular forms and vector-valued
k,χ,ρ k,χ,ρ
cusp forms, respectively, for which we require that each a (n) = 0 when n + κ is negative,
j,γ j,γ
respectively, non-positive.
2.2. Vector-valued Eichler integrals. Inthissubsection, weintroducethevector-valuedEichler
integrals. Let P be the vector space of vector-valued functions G(τ) = p G (τ)e holomorphic
k j=1 j j
in H such that G (τ) is a polynomial of degree at most k. It is worth mentioning that the space P
j k
P
is preserved under the slash operator. With this underlying space, we introduce the vector-valued
Eichler integrals.
Definition 2.2. Let ρ be a p-dimensional representation ρ : Γ GL(p,C), k an arbitrary non-
→
negative integer and χ a character of Γ. A vector-valued Eichler integral of weight k, character
−
χ and type ρ on Γ is a vector-valued function F(τ) on H satisfying
(F γ)(τ) F(τ) P
−k,χ,ρ k
| − ∈
for all γ Γ.
∈
4 DOHOONCHOI ANDSUBONGLIM
If we let p (τ) = (F γ)(τ) F(τ), then the vector-valued functions p (τ) are called period
γ −k,χ,ρ γ
| −
functions of F(τ). Then it turns out that p γ Γ satisfies the following cocycle condition: for
γ
{ | ∈ }
γ ,γ Γ,
1 2
∈
(2.1) p (τ) = p (τ)+(p γ )(τ).
γ1γ2 γ2 γ1|−k,χ,ρ 2
A collection p γ Γ of elements of P satisfying (2.1) is called a cocycle and a coboundary is
γ k
{ | ∈ }
a collection p γ Γ such that
γ
{ | ∈ }
p (τ) = (p γ)(τ) p(τ)
γ −k,χ,ρ
| −
for γ Γ with a fixed element p(τ) P . A parabolic cocycle p γ Γ is a collection of
k γ
∈ ∈ { | ∈ }
elements of P satisfying (2.1), in which for every parabolic class in Γ there exists a fixed
k
B
element Q (τ) P such that
B k
∈
p (τ) = (Q B)(τ) Q (τ),
B B −k,χ,ρ B
| −
for all B .
∈ B
A vector-valued generalized Poincar´e series gives an example of the vector-valued Eichler in-
tegrals. A vector-valued generalized Poincar´e series was first defined by Lehner [23]. We recall
the definition of a vector-valued generalized Poincar´e series. Let g γ Γ be a parabolic co-
γ
{ | ∈ }
cycle of elements of P of weight k Z with χ a multiplier system in Γ of weight k and ρ a
k
− ∈ −
representation. Assume also that g (τ) = 0, where
Q
(2.2) Q = (1 λ) Γ, λ > 0,
0 1 ∈
is a generator of Γ . We define a vector-valued generalized Poincar´e series as
∞
p
(g ) (τ)
(2.3) Φ(τ;r) = V j e ,
(cτ +d)r j
Xj=1 V=(Xa b)∈L
c d
where r is a large positive even integer and is any set in Γ containing all transformations with
L
different lower rows.
Theorem 2.3. [4, Theorem 3.6] The generalized Poincar´e series Φ(τ;r), defined as in (2.3),
converges absolutely and uniformly on compact subsets of H for sufficiently large r.
ItwasprovedusingageneralizedPoincar´eseriesΦ(τ;r)theexistenceofthevector-valuedEichler
integral forgiven periodfunctions in[4]. Following theliteratureontheEichler cohomologytheory,
we introduce a left-finite expansion at each parabolic cusp for the consistency. The expansion at
the cusp i has the form
∞
p
F(τ) = a (n,j)e2πi(n+κj,0)/λ0e ,
0 j
j=1 n≫−∞
X X
where Q = Q0 = 01 λ10 , λ0 > 0, is a generator of Γ∞ and
(cid:0) (cid:1) e2πiκ1,0
·
(2.4) χ(Q)ρ(Q) = · .
·
!
e2πiκp,0
Let q , ,q be the inequivalent parabolic cusps other than infinity. Suppose also that Q =
1 t i
∗ ∗ ···
( ), 1 i t, is a parabolic generator, i.e., Q is a generator of Γ , where Γ is the cyclic
ci di ≤ ≤ i j j
COHOMOLOGICAL RELATION BETWEEN JACOBI FORMS AND SKEW-HOLOMORPHIC JACOBI FORMS 5
subgroup of Γ fixing q , 1 i t. To describe the expansion at a finite parabolic cusp q , choose
i i
A := 0 −1 , so that A h≤as d≤eterminant 1 and A (q ) = . Then the width of the cusp q is a
i 1 −qi i i i ∞ i
positive real number such that A−1 1 λi A = Q . The expansion at the cusp q has the form
(cid:0) (cid:1) i 0 1 i i i
(cid:0) (cid:1) p −2πi(n+κj,i)
F(τ) = (τ qi)k ai(n,j)e λi(τ−qi) ej,
−
j=1 n≫−∞
X X
where
e2πiκ1,i
·
(2.5) χ(Q )ρ(Q ) = · ,
i i
·
!
e2πiκp,i
for 0 κ < 1, 1 j p and 1 i t. (The motivation for the form of these expansions can
j,i
≤ ≤ ≤ ≤ ≤
be found in [20, pp. 17-20].) With these notations the following theorem gives the existence of the
vector-valued Eichler integral for given period functions.
Theorem 2.4. [4, Theorem 3.9] Assume that k is a nonnegative integer. Let χ be a character of Γ
and ρ a unitary representation. Suppose g γ Γ is a parabolic cocycle of weight k, character
γ
{ | ∈ } −
χ and type ρ on Γ in P . Then there exists a vector-valued function Φ(τ), holomorphic in H, such
k
that
(Φ γ)(τ) Φ(τ) = g (τ)
−k,χ,ρ γ
| −
for all γ Γ.
∈
Remark 2.5. Since g γ Γ is a parabolic cocycle, there is an element g (τ) P such that
γ i k
{ | ∈ } ∈
g (τ) = (g Q )(τ) g (τ),
Qi i|−k,ν,ρ i − i
for 0 i t. Then Φ(τ) g (τ) is invarinat under the slash operator Q and hence has the
i −k,χ,ρ i
≤ ≤ − |
expansions at parabolic cusps q , 0 i t, of the forms
i
≤ ≤
p
−2πi(n+κj,i)
Φ(τ) = gi(τ)+(τ qi)k ai(n,j)e λi(τ−qi) ej, 1 i t,
− ≤ ≤
j=1 n≫−∞
X X
p
2πi(n+κj,0)
Φ(τ) = g0(τ)+ a0(n,j)e λ0 ej, i = 0.
j=1 n≫−∞
X X
2.3. Vector-valued Poincar´e series and supplementary functions. In this subsection we
review the theory of Poincar´e series for vector-valued modular forms and introduce the supple-
mentary function theory, following [3].
Definition 2.6. Fix integers n and α with 1 α p. The Poincar´e series P (τ) is defined
n,α,χ,ρ
≤ ≤
as
1 e2πi(−n+κα)γτ/λ
(2.6) P (τ) := ρ(γ)−1e ,
n,α,χ,ρ 2 χ(γ)(cτ +d)k α
γ=X(a b)
c d
where γ = (a b) ranges over a set of coset representatives for < Q > Γ and κ = κ is as in
c d \ α α,0
(2.4).
6 DOHOONCHOI ANDSUBONGLIM
The series (2.6) is well-defined and invariant with respect to the action of Γ if we assume
k,χ,ρ
|
the absolute convergence of the series P (τ). It is known that if k > 2 then the component
n,α,χ,ρ
function (P ) (τ) of P (τ), 1 j p, converges absolutely uniformly on compact subsets
n,α,χ,ρ j n,α,χ,ρ
≤ ≤
of H. In paricular, each (P ) (τ) is holomorphic in H (see Proposition 2.5 in [3]) and its
n,α,χ,ρ j
Fourier expansion at i is of the form (see [3, Theorem 2.6])
∞
p p
(2.7) (P ) (τ) = δ e2πi(−n+κα)τ/λ + a(l,j)e2πi(l+κj)τ/λ.
n,α,χ,ρ j α,j
Xj=1 Xj=1 l+Xκj>0
Now we look at the properties of the Poincar´e series P (τ). One can see that the function
n,α,χ,ρ
P (τ) vanishes at all cusps of Γ which are not equivalent to i and that S (Γ) is spanned
n,α,χ,ρ k,χ,ρ
∞
by Poincar´e series P (τ) with n+κ > 0 (see Theorem 2.7 in [3]).
n,α,χ,ρ α
Suppose f(τ) S (Γ) with k > 0. Then there exist complex numbers b , ,b such that
k+2,χ,ρ 1 s
∈ ···
f(τ) = s b P (τ). Put f∗(τ) = s b P (τ), where
i=1 i ni,αi,χ,ρ i=1 i n′i,αi,χ¯,ρ¯
P P n if κ = 0,
′ i α
n = −
i 1 n if κ > 0.
( i α
−
Note that if we let
e2πiκ′1
·
χ¯(Q)ρ¯(Q) = · ,
 · 
e2πiκ′p
with 0 κ′ < 1 for 1 j p, then we have  
≤ j ≤ ≤
0 if κ = 0,
′ j
κ =
j 1 κ if κ > 0,
( j
−
for 1 j p. Thus we have the expansion at i
≤ ≤ ∞
p
Pn′i,αi,χ¯,ρ¯(τ) = e2πi(−n′i+κ′αi)τ/λeαi + an′i,αi(l,j)e2πi(l+κj)τ/λej
Xj=1 l+Xκj>0
p
= e2πi(ni−καi)τ/λeαi + an′i,αi(l,j)e2πi(l+κj)τ/λej.
Xj=1 l+Xκj>0
It follows that f∗(τ) M! (Γ), f∗(τ) has a pole at i with principal part
∈ k+2,χ¯,ρ¯ ∞
s
bie2πi(ni−καi)τ/λ,
i=1
X
and f∗(τ) vanishes at all of the other cusps of Γ. We call f∗(τ) the function supplementary to
f(τ).
Functions f(τ) and f∗(τ) have important relations which can be expressed in terms of period
functions. A form f(τ) M! (Γ) is a vector-valued weakly holomorphic cusp form if its
∈ k+2,χ,ρ
constant term vanishes. Let S! (Γ) denote the space of vector-valued weakly holomorphic
k+2,χ,ρ
cusp forms.
COHOMOLOGICAL RELATION BETWEEN JACOBI FORMS AND SKEW-HOLOMORPHIC JACOBI FORMS 7
Suppose that f(τ) S (Γ). Then we have a nonholomorphic Eichler integral of f(τ)
k+2,χ,ρ
∈
1 i∞ −
N(τ) := f(z)(τ¯ z)kdz ,
Ef c −
k+2(cid:20)Zτ (cid:21)
where c = (k−2)! and [ ]− indicates the complex conjugate of the function inside [ ]−. On the
k −(2πi)k−1
p
other hand, suppose that f(τ) = a(n,j)e2πi(n+κj)τ/λe S! (Γ). Then we have a
j ∈ k+2,χ,ρ
j=1 n≫−∞
X n+Xκj6=0
holomorphic Eichler integral of f(τ)
p −(k+1)
n+κ
H(τ) := a(n,j) j e2πi(n+κj)τ/λe +c .
Ef λ j f
j=1 n≫−∞ (cid:18) (cid:19)
X n+Xκj6=0
Here, the constant term c is determined by the fact that the Fourier coefficients of modular forms
f
of negative weight are completely determined by the principal part of the expansion of those forms
at the cusps (see [21]). For example, if we assume that f(τ) has a pole at i and that it is
∞
holomorphic at all other cusps, then c is equal to
f
p p k+2
1 2πi
cf := δκj,0 λ(k +1)! a(l,t) −c χ−1(γ)ρ(γ−1)j,te2cπλi(l+κt)a ej.
Xj=1 (cid:18) Xt=1 Xl<0 γ=(aXb)∈C+ (cid:18) (cid:19) (cid:19)
c d
We introduce the period functions for f(τ) by
rH(f,γ;τ) := c ( H H γ)(τ),
k+2 Ef −Ef |−k,χ,ρ
rN(f,γ;τ) := c ( N N γ)(τ),
k+2 Ef −Ef |−k,χ¯,ρ¯
where γ Γ. For the period functions we have the following theorem.
∈
Theorem 2.7. [3, Theorem 2.8], [14, Section 2] Suppose that k is a positive integer and f(τ)
∈
S (Γ). Then we have
k+2,χ,ρ
(1) rH(f,γ;τ) = [rH(f∗,γ;τ¯)]− for all γ Γ,
(2) rH(f,γ;τ) = [rN(f,γ;τ¯)]− for all γ ∈Γ.
∈
3. Jacobi forms and associated cohomology groups
In this section, we review basic notions of Jacobi forms (see [10, 27]) and skew-holomorphic
Jacobi forms (see [24, 25]). We also review the definition of Jacobi integrals (see [7]) and define
the cohomology group for Jacobi forms of half-integral weight.
3.1. Jacobi forms. Firstwefixsomenotations. LetΓJ = Γ⋉Z2 beaJacobi groupwithassociated
composition law
(γ ,(λ ,µ )) (γ ,(λ ,µ )) = (γ γ ,(λ +λ ,µ +µ )),
1 1 1 2 2 2 1 2 1 2 1 2
·
where (λ,µ) = (λ,µ) γ . Then ΓJ acts on H C as a group of automorphism. The action is given
2
· × e
by e
e z +λτ +µ
e (γ,(λ,µ)) (τ,z) = γτ, ,
· cτ +d
(cid:18) (cid:19)
8 DOHOONCHOI ANDSUBONGLIM
where γτ = aτ+b for γ = (a b) Γ. Let k be a half-integer and χ be a multiplier system of weight
cτ+d c d ∈
k on Γ, i.e., χ : Γ C satisfies
→
(1) χ(γ) = 1 for all γ Γ,
| | ∈
(2) χ satisfies the consistency condition
χ(γ )(c τ +d )k = χ(γ )χ(γ )(c γ τ +d )k(c τ +d )k,
3 3 3 1 2 1 2 1 2 2
∗ ∗
where γ = γ γ and γ = ( ),i = 1,2 and 3,
3 1 2 i ci di
(3) χ satisfies χ( I) = eπik.
−
For γ = (a b) Γ,X = (λ,µ) Z2 and m Z with m > 0, we define
c d ∈ ∈ ∈
(Φ k,m,χγ)(τ,z) := (cτ +d)−kχ¯(γ)e−2πimccτz+2dΦ(γ(τ,z))
|
and
(Φ X)(τ,z) := e2πim(λ2τ+2λz+µλ)Φ(τ,z +λτ +µ),
m
|
where γ(τ,z) = (aτ+b, z ). Then ΓJ acts on the space of functions on H C by
cτ+d cτ+d ×
(3.1) (Φ (γ,X))(τ,z) := (Φ γ X)(τ,z).
k,m,χ k,m,χ m
| | |
We introduce the definition of a Jacobi form.
Definition 3.1. A Jacobi form of weight k, index m and multiplier system χ on ΓJ is a holomor-
phic mapping Φ(τ,z) on H C satisfying
×
(1) (Φ γ)(τ,z) = Φ(τ,z) for every γ Γ,
k,m,χ
| ∈
(2) (Φ X)(τ,z) = Φ(τ,z) for every X Z2,
m
| ∈
(3) for each γ = (a b) SL(2,Z), the function (cτ + d)−ke2πim−cτc+z2dΦ((γ,0) (τ,z)) has the
c d ∈ ·
Fourier expansion of the form
(3.2) a(l,r)e2πi(l+κγ)/λγe2πirz,
l,r∈Z
4(l+κγ)X−λγmr2≥0
with a suitable 0 κ < 1, λ Z.
γ γ
≤ ∈
If a Jacobi form satisfies the condition a(l,r) = 0 only if 4(l +κ ) λ mr2 > 0, then it is called
γ γ
6 −
a Jacobi cusp form.
Now we look into the theta series, which plays an important role in the proofs of our main
theorems. Let S be a positive integer and a,b Q. We consider the theta series
∈
θ (τ,z) := eπiS((λ+a)2τ+2(λ+a)(z+b))
S,a,b
λ∈Z
X
with characteristic (a,b) converging normally on H C.
×
We are in a position to explain the beautiful theory of the theta expansion, developed by Eichler
and Zagier [10], which gives an isomorphism between (skew-holomorphic) Jacobi forms and vector-
valued modular forms.
COHOMOLOGICAL RELATION BETWEEN JACOBI FORMS AND SKEW-HOLOMORPHIC JACOBI FORMS 9
Theorem 3.2. [27, Section 3] Let Φ(τ,z) be holomorphic as a function of z and satisfy
(3.3) (Φ X)(τ,z) = Φ(τ,z) for every X Z2.
m
| ∈
Then we have
(3.4) Φ(τ,z) = f (τ)θ (τ,z)
a 2m,a,0
a∈N
X
with uniquely determined holomorphic functions f : H C, where = Z/2mZ. If Φ(τ,z) also
a
→ N
satisfies the transformation
(Φ γ)(τ,z) = Φ(τ,z) for every γ SL(2,Z),
k,m,χ
| ∈
then we have for each a
∈ N
−j
fa −τ1 = χ((10 −01))det τi 2τk(2m)−12 e2πi(2mab)fb(τ)
(cid:18) (cid:19) (cid:18) (cid:19) b∈N
X
and
f (τ +1) = χ((1 1))e−2πima2f (τ).
a 0 1 a
Furthermore, if Φ(τ,z) is a Jacobi form in J (Γ(1,j)), then functions in f a necessarily
k,m,χ a
{ | ∈ N}
must have the Fourier expansions of the form
f (τ) = a(l)e2πilτ.
a
l≥0
raXtional
The decomposition by theta functions as in (3.4) is called the theta expansion. Now we explain
the isomorphism between Jacobi forms and vector-valued modular forms induced by the theta
expansion more precisely. First, we can define the space of vector-valued modular forms associated
with J (ΓJ) by the theta expansion. Let Φ(τ,z) be a Jacobi form in J (ΓJ). By Theorem
k,m,χ k,m,χ
3.2, we have the theta expansion
Φ(τ,z) = f (τ)θ (τ,z).
a 2m,a,0
a∈N
X
We take a multiplier system χ′′ of weight 1/2 on SL(2,Z), for example we can take a power of
eta-multiplier system: χ′′(γ) = η(γτ) for γ SL(2,Z), where η(τ) = eπiτ ∞ (1 e2πinτ) is the
12
η(τ) ∈ n=1 −
Dedekind eta function. Then we define a representation ρ′ : SL(2,Z) GL( ,C) by
→ Q|N|
(3.5) ρ′(T)e = χ′′(T)e−2πima2e
a a
and
1
i
(3.6) ρ′(S)e = χ′′(S) 2 e2πi(2mab)e ,
a b
√2m
b∈N
X
where T = (1 1) and S = (0 −1). Let χ′ be a character of Γ defined by
0 1 1 0
(3.7) χ′(γ) = χ(γ)χ′′(γ).
Then a vector-valued function f (τ)e is a vector-valued modular form in M (Γ).
a∈N a a k−1,χ′,ρ′
2
P
10 DOHOONCHOI ANDSUBONGLIM
Theorem 3.3. [27, Theorem 3.3] The theta expansion gives an isomorphism between J (ΓJ)
k,m,χ
and M (Γ). Furthermore, this isomorphism sends Jacobi cusp forms to vector-valued cusp
k−1,χ′,ρ′
2
forms.
3.2. Skew-holomorphic Jacobi forms. Skoruppa [24, 25] introduced skew-holomorphic Jacobi
forms, which play a crucial role in understanding liftings of modular forms and Jacobi forms. Now
we briefly discuss the definition of skew-holomorphic Jacobi forms. For a half-integer k and a
integer m, we have the following slash operator on functions φ : H C C:
× →
(φ sk (γ,X))(τ,z) := χ¯(γ)(cτ¯+d)1−k cτ +d −1e2πim(−c(z+cλτ+τ+dµ)2+λ2τ+2λz)φ((γ,X) (τ,z)),
|k,m,χ | | ·
for all (γ,X) = (a b),(λ,µ) ΓJ, where χ is a multiplier system of weight k on Γ.
c d ∈
(cid:18) (cid:19)
Definition 3.4. A function φ : H C C is a skew-holomorphic Jacobi form of weight k, index
× →
m and multiplier system χ on ΓJ if φ(τ,z) is real-analytic in τ H, is holomorphic in z C, and
∈ ∈
satisfies the following conditions:
(1) for all (γ,X) ΓJ, (φ sk (γ,X))(τ,z) = φ(τ,z),
∈ |k,m,χ
(2) for each γ = (a b) SL(2,Z), the function (cτ +d)−ke2πim(−ccτz+2d)Φ((γ,0) (τ,z)) has the
c d ∈ ·
Fourier expansion of the form
−π(λγr2−4m(l+κγ))v
(3.8) a(l,r)e mλγ e2πiτ(l+κγ)/λγe2πirz,
l,r∈Z
4m(l+κXγ)−λγr2≤0
with a suitable 0 κ < 1, λ Z.
γ γ
≤ ∈
If the Fourier expansion in (3.8) is only over 4m(l + κ ) λ r2 < 0, then φ(τ,z) is a skew-
γ γ
−
holomorphic Jacobi cusp form of weight k, index m and multiplier system χ on ΓJ. We denote the
spaces of skew-holomorphic Jacobi forms and skew-holomorphic Jacobi cusp forms, each of weight
k, index m and multiplier system χ on ΓJ, by Jsk (ΓJ) and Jsk,cusp(ΓJ), respectively.
k,m,χ k,m,χ
Bringmann and Richter in [2] found the close relationship between skew-holomorphic Jacobi
forms and vector-valued modular forms. Like a Jacobi form, a skew-holomorphic Jacobi form has
the theta decomposition. More precisely, if φ(τ,z) Jsk (ΓJ), then we have
∈ k,m,χ
φ(τ,z) = f (τ)θ (τ,z).
a 2m,a,0
a∈N
X
But in this case, f (τ)e is a vector-valued modular form in M (Γ) where χ′ and ρ′ are
a∈N a a k−1,χ′,ρ′
2
definedin(3.7)and(3.5), (3.6), respectively. Thisgivesanisomorphismbetweenskew-holomorphic
P
Jacobi forms and vector-valued modular forms.
Theorem 3.5. [2, Section 6] The theta expansion gives an isomorphism between Jsk (ΓJ) and
k,m,χ
M (Γ). Furthermore, this isomorphism sends skew-holomorphic Jacobi cusp forms to vector-
k−1,χ′,ρ′
2
valued cusp forms.