On liftings of modular forms and Weil representations

Fredrik Strömberg

doi:10.1515/forum-2022-0353

Article Open Access

On liftings of modular forms and Weil representations

Fredrik Strömberg

Published/Copyright: August 31, 2023

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From the journal Forum Mathematicum Volume 36 Issue 1

Abstract

We give an explicit construction of lifting maps from integral and half-integral modular forms to vector-valued modular forms for Weil representations associated with arbitrary isotropic subgroups and finite quadratic modules of even and odd signature. This construction provides an explicit and general extension of previous work by Scheithauer and Zhang.

Keywords: Modular forms; Weil representations; finite quadratic modules; liftings

MSC 2020: 11F03; 11F11; 11F06; 11F37; 11F27

1 Introduction

Vector-valued modular forms for Weil representations play a fundamental role in the construction of automorphic products by Borcherds and others; cf., e.g., [1, 13, 2]. In addition to this connection with automorphic forms on orthogonal groups, the Weil representation also provides a unified framework for working with modular forms of arbitrary level and integral and half-integral weight. It can indeed be shown that every irreducible representation of Mp 2 ⁡ ( ℤ ) whose kernel is a congruence subgroup is isomorphic to an irreducible constituent of a Weil representation [16]. This isomorphism in terms of irreducible representations is non-constructive on the level of individual spaces of modular forms, but there has been some progress for certain symmetric subspaces by, e.g., Scheithauer [13].

The main result of this paper is an explicit construction of lifting maps which takes integral or half-integral weight classical modular forms to vector-valued modular forms for the associated Weil representation. This type of maps was first considered in certain cases by Schoeneberg [15] and later by Oda [11], Bundschuh [4] and Bruinier and Bundschuh [3]. The first more general formal treatment was given by Scheithauer [12, 13, 14] in the case of integral weight modular forms, corresponding to finite quadratic modules of even signature. Some specific cases of half-integral weight modular forms and finite quadratic modules of odd signature were recently considered by Zhang [20, 21]. The injectivity of the lifting maps in different settings has also been discussed by the previously mentioned authors, where explicit domains and codomains for isomorphisms have been introduced.

With notation to be introduced later, our main results are combined in the following theorem.

Theorem (Theorem 5.6 and 5.8).

Let D be a finite quadratic module, let k be an integer or half-integer, let F ∈ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) , let S 0 ⩽ D be an isotropic subgroup, and define

Ψ S 0 ⁢ ( F ) ⁢ ( τ ) = ∑ M ∈ Γ 0 ⁢ ( N ) ∖ SL 2 ⁡ ( ℤ ) 𝐜 M ⁢ F | k ⁢ M ⁢ ( τ ) ,

where

𝐜 M = ρ D ⁢ ( M ) - 1 ⁢ ∑ γ ∈ S 0 𝐞 γ .

Then Ψ S 0 ⁢ ( F ) ∈ M k ⁢ ( ρ D ) and if F is a cusp form, then so is Ψ S 0 ⁢ ( F ) . Furthermore, Ψ S 0 ⁢ ( F ) is invariant under all automorphisms of D which stabilize S 0 and if we write Ψ S 0 ⁢ ( F ) = ∑ α ∈ D f α ⁢ e α , then

f α ⁢ ( τ ) = ∑ n ∈ ℤ + Q ⁢ ( α ) c ⁢ ( α , n ) ⁢ q n ,

with

c ⁢ ( α , n ) = ∑ i = 1 r Λ i ⁢ ( α ) ⁢ c i ⁢ ( n ⁢ w i ) , n ∈ ℤ + Q ⁢ ( α ) ,

where the sum is taken over a complete set of cusp representatives, a 1 , … , a r , for Γ 0 ⁢ ( N ) , the constants Λ i ⁢ ( α ) are explicitly given and c i ( n w i ) is the Fourier coefficient of q n for F at the cusp a i .

To prove these theorems, we use a similar approach to Scheithauer [12, 13] together with a careful study of the additional cases arising for finite quadratic modules of odd signature to include half-integral weight modular forms. A crucial element in this is the explicit formulas for the Weil representation given by the author in [17].

When comparing our results and proofs with that of [12, 13], it is important to note that the Weil representation defined here is the dual of the one used there. We also work directly with explicit indecomposable modules as Jordan components instead of the genus symbols formalism. We prefer this approach because it is more tractable for computations and the potential non-uniqueness of 2-adic symbols as described in, e.g., [6, Section 15.7.5].

An important general observation is that all of the lifting maps preserve regularity conditions and we will therefore, for simplicity, give all formulations in terms of holomorphic modular forms.

In Section 2, we will provide a brief background to finite quadratic modules and their Jordan decompositions, the metaplectic group, multiplier systems and cusps of Γ 0 ⁢ ( N ) and their stabilizers. In Section 3 we will define the Weil representation. In Section 4, we briefly introduce scalar and vector-valued modular forms and then present the main results and their proofs in Section 5 before discussing injectivity in Section 6. Finally, we provide explicit examples in Section 7.

2 Background

Let us first introduce some standard notation and definitions. Let ℍ = { τ = x + i ⁢ y ∣ y > 0 } be the complex upper half-plane. For A = ( a b c d ) ∈ SL 2 ⁡ ( ℝ ) and τ ∈ ℍ , let

A ⁢ τ = a ⁢ τ + b c ⁢ τ + d , j A ⁢ ( τ ) = c ⁢ τ + d

and write j A ⁢ ( τ ) 1 / 2 = c ⁢ τ + d , where we choose the principal branch of the square-root. More generally, for any z ∈ ℂ and k ∈ ℝ , we always define z k = e k ⁢ Log ⁡ ( z ) with Log ⁡ ( z ) = ln ⁡ | z | + i ⁢ Arg ⁡ ( z ) , where ln denotes the natural logarithm on ℝ + and Arg ( z ) ∈ ] - π , π ] is the principal branch. For z ∈ ℂ , we write e ⁢ ( z ) = e 2 ⁢ π ⁢ i ⁢ z and if n ∈ ℕ , we set e n ⁢ ( z ) = e ⁢ ( z / n ) . We let Γ = SL 2 ⁡ ( ℤ ) denote the modular group. For a positive integer N, the principal congruence subgroup Γ ⁢ ( N ) and the congruence subgroups

Γ ⁢ ( N ) ⩽ Γ 1 ⁢ ( N ) ⩽ Γ 0 ⁢ ( N ) ⩽ SL 2 ⁡ ( ℤ )

are defined by

Γ ⁢ ( N ) = { ( a b c d ) ∈ SL 2 ⁡ ( ℤ ) | a - 1 ≡ d - 1 ≡ b ≡ c ≡ 0 ( mod N ) } ,

Γ 1 ⁢ ( N ) = { ( a b c d ) ∈ SL 2 ⁡ ( ℤ ) | a - 1 ≡ d - 1 ≡ c ≡ 0 ( mod N ) } ,

Γ 0 ⁢ ( N ) = { ( a b c d ) ∈ SL 2 ⁡ ( ℤ ) | c ≡ 0 ( mod N ) } .

2.1 Lattices and finite quadratic modules

Although the abstract theory of finite quadratic modules is sufficient for our discussions, it might be helpful to introduce the topic via lattices and their discriminant forms as these are canonical examples.

Let V be a rational vector space with a non-degenerate symmetric bilinear form b : V × V → ℚ and a quadratic form q : V → V related through the polarization identity

q ⁢ ( x + y ) = q ⁢ ( x ) + q ⁢ ( y ) + b ⁢ ( x , y ) ,

or equivalently b ⁢ ( x , x ) = 2 ⁢ q ⁢ ( x ) . A lattice in V is a finitely generated ℤ -module of full rank together with the bilinear and quadratic forms. The signature of the lattice L is defined by sign ⁡ ( L ) = r + - r - , where r + and r - are the numbers of positive and negative eigenvalues of the Gram matrix of L.

Let L be an even integral lattice in V, in other words, q ⁢ ( L ) ⊂ ℤ , b ⁢ ( L , L ) ⊂ ℤ and b ⁢ ( x , x ) ∈ 2 ⁢ ℤ for all x ∈ L . The dual lattice of L is defined by

L # = { x ∈ V ∣ b ⁢ ( x , y ) ∈ ℤ ⁢ for all ⁢ y ∈ L } ,

and it is easy to see that L is contained in L # with finite index. The quotient L # / L is a finite abelian group called the discriminant group of L. The discriminant form of L, that is, disc ⁡ ( L ) = ( L # / L , Q ) , consists of the discriminant group together with the quadratic form Q : L # / L → ℚ / ℤ defined by Q ⁢ ( x + L ) = q ⁢ ( x ) + ℤ . The associated bilinear form

B : L # / L × L # / L → ℚ / ℤ

is again determined by

B ⁢ ( x , y ) = Q ⁢ ( x + y ) - Q ⁢ ( x ) - Q ⁢ ( y ) .

The discriminant form is an example of a finite quadratic module ( D , Q ) , where D is a finite abelian group and Q : D → ℚ / ℤ is a quadratic form with associated non-degenerate symmetric bilinear form given by

B ⁢ ( x , y ) = Q ⁢ ( x + y ) - Q ⁢ ( x ) - Q ⁢ ( y ) .

We often use D to denote both the abelian group and the finite quadratic module, with the precise meaning being clear from the context. The level of D is the smallest positive integer N such that N ⁢ Q ⁢ ( x ) ∈ ℤ for all x ∈ D . It is possible to define the signature of D by sign ⁡ ( D ) ∈ ℤ / 8 ⁢ ℤ by Milgram’s formula

e 8 ⁢ ( sign ⁡ ( D ) ) = 1 | D | ⁢ ∑ γ ∈ D e ⁢ ( Q ⁢ ( γ ) ) ,

and if D = disc ⁡ ( L ) , then sign ⁡ ( D ) ≡ sign ⁡ ( L ) ( mod 8 ) . It is not hard to see that N and | D | have the same prime divisors. More precisely, if sign ⁡ ( D ) is even, then the level N divides | D | , and if sign ⁡ ( D ) is odd, then | D | is even and N divides 2 ⁢ | D | , in particular 4 ∣ N ; cf., e.g., [5, Remark 14.3.23].

The Weil representation is naturally associated with finite quadratic modules, and we therefore prefer to use this formulation, but it is known that every finite quadratic module is isomorphic to the discriminant form of a lattice [19]. However, the discriminant form only determines the lattice up to a so-called stable isomorphism, that is, up to the addition of unimodular lattices [9].

2.2 Jordan decompositions and subgroups

If ( D , Q ) is a finite quadratic module, it has an orthogonal decomposition corresponding to the Sylow decomposition of the finite abelian group D. This is said to be a Jordan decomposition of D and is in general not unique for the prime 2. A common approach is to describe Jordan decompositions in terms of so-called genus symbols. This is used in, e.g., [13, 6]. Here we will instead follow the approach introduced in [16] and work directly with explicit indecomposable modules of the following types. For a prime p, an integer k ≥ 1 and t not divisible by p, we define

A p k t = ( ℤ / p k ℤ , x ↦ t x 2 / p k + ℤ ) for ⁢ p > 2 ,

A 2 k t = ( ℤ / 2 k ℤ , x ↦ t x 2 / 2 k + 1 + ℤ ) for ⁢ p = 2 ,

B 2 k = ( ℤ / 2 k ℤ ⊕ ℤ / 2 k ℤ 2 , ( x , y ) ↦ ( x 2 + x y + y 2 ) / 2 k + ℤ ) for ⁢ p = 2 ,

C 2 k = ( ℤ / 2 k ℤ ⊕ ℤ / 2 k ℤ 2 , ( x , y ) ↦ x y / 2 k + ℤ ) for ⁢ p = 2 .

It is easy to see that the level of A p k t is p k for p > 2 , the level of A 2 k t is 2 k + 1 , and the levels of B 2 k and C 2 k are both 2 k . It is also easy to calculate the signatures in ℤ / 8 ⁢ ℤ of these modules; cf., e.g., [5, Chapter 14.5.2]. The signature of A p k t is 1 - p k if p > 2 , and t or t + 4 depending on whether or not ( t 2 ) k = 1 or -1 if p = 2 . The signature of B 2 k is 0 if k is even, and 4 if k is odd, and finally the signature of C 2 k is 0.

For our purposes, a Jordan decomposition of D is simply an orthogonal decomposition into a finite number of indecomposable modules, called Jordan components. For a prime p, by D ⁢ ( p ) we mean the direct sum of all constituents of p-power order, and D ⁢ [ p k ] denotes the direct sum of all constituents of order exactly p k .

If S ⩽ D is a subgroup, we note that ( S , Q ) is also a finite quadratic module, and if

S ⊥ = { α ∈ D ∣ B ⁢ ( α , γ ) = 0 ⁢ for all ⁢ γ ∈ S }

is the orthogonal complement of S, then D = S ⊕ S ⊥ . An element γ ∈ D is said to be isotropic if Q ⁢ ( γ ) = 0 , and a subgroup S 0 ⩽ D is said to be isotropic if all its elements are isotropic.

Let c be an integer and define D c and D c as the kernel and image in D of multiplication by c. Then D = D c ⊕ D c and if we define

D c ⁣ * = { α ∈ D ∣ c ⁢ Q ⁢ ( γ ) + B ⁢ ( γ , α ) ∈ ℤ ⁢ for all ⁢ γ ∈ D c } ,

then D c ⁣ * / D c = { x c } for an explicit element x c ∈ D and the map Q c : D c ⁣ * → ℚ / ℤ defined by

Q c ⁢ ( x c + y ⁢ c ) = c ⁢ Q ⁢ ( y ) + B ⁢ ( x c , y ) + ℤ

is well-defined and satisfies (cf., e.g., [17, Lemma 2.5])

c ⁢ Q c ⁢ ( x c + y ⁢ c ) ≡ Q ⁢ ( x c + y ⁢ c ) - Q ⁢ ( x c ) + ℤ .

If S 0 is an isotropic subgroup, then D = c - 1 ⁢ S 0 ⊕ c ⁢ S 0 ⊥ and if we define

D S 0 c ⁣ * = { α ∈ D ∣ c ⁢ Q ⁢ ( γ ) + B ⁢ ( γ , α ) ∈ ℤ ⁢ for all ⁢ γ ∈ c - 1 ⁢ S 0 ∩ S 0 ⊥ } ,

then D S 0 c ⁣ * / ( S 0 + c ⁢ S 0 ⊥ ) = { x c * } for some x c * ∈ D . For more details and proofs, see [13].

2.3 The metaplectic group

The real metaplectic group Mp 2 ⁡ ( ℝ ) is a central extension of degree two of SL 2 ⁡ ( ℝ ) and can be defined as a set of pairs ( A , φ A ) , where A ∈ SL 2 ⁡ ( ℝ ) and φ A : ℍ → ℂ is a holomorphic function that satisfies φ A ⁢ ( τ ) 2 = j A ⁢ ( τ ) , in other words, φ A ⁢ ( τ ) = ± j A ⁢ ( τ ) 1 / 2 , and with a group law given by

( A , φ A ) ( B , φ b ) = ( A B , τ ↦ φ A ( B τ ) φ B ( τ ) ) .

For the purpose of this paper, we will only consider Mp 2 ( ℤ ) = π - 1 ( SL 2 ( ℤ ) ), where π : ( A , φ A ) ↦ A is the natural projection. It is often convenient to use the section s : SL 2 ⁡ ( ℝ ) → Mp 2 ⁡ ( ℝ ) defined by s ⁢ ( A ) = ( A , j A ⁢ ( z ) 1 / 2 ) for A ∈ SL 2 ⁡ ( ℝ ) . The group law is then given by

s ⁢ ( A ⁢ B ) = σ 1 / 2 ⁢ ( A , B ) ⁢ s ⁢ ( A ) ⁢ s ⁢ ( B ) ,

where, for k ∈ 1 2 ⁢ ℤ ,

σ k : Γ × Γ → { 1 , e ± 2 ⁢ π ⁢ i ⁢ k }

is a 2-cocycle, explicitly given by

σ k ⁢ ( A , B ) = j A ⁢ ( B ⁢ τ ) k ⁢ j B ⁢ ( τ ) k ⁢ j A ⁢ B ⁢ ( τ ) - k .

It is easy to see that Mp 2 ⁡ ( ℤ ) is generated by the elements

T = ( ( 1 1 0 1 ) , 1 ) and S = ( ( 0 - 1 1 0 ) , τ ) ,

and the center of Mp 2 ⁡ ( ℤ ) is generated by the element Z = S 2 = ( - I 2 , i ) of order 4, where I 2 is the identity 2-by-2 matrix.

2.4 Multiplier systems

Let k be an integer or half-integer. A multiplier system of weight k on a subgroup G ⩽ Γ is a function v : G → ℂ * such that

v ⁢ ( A ⁢ B ) = v ⁢ ( A ) ⁢ v ⁢ ( B ) ⁢ σ k ⁢ ( A , B ) for all ⁢ A , B ∈ G .

By setting A = B = Z , it is easy to see that v must satisfy v ⁢ ( Z ) = e - π ⁢ i ⁢ k . A multiplier system of integral weight is simply a character on G, and a multiplier system of half-integral weight is a character of the inverse image of G in Mp 2 ⁡ ( ℤ ) .

Let ρ : Γ → GL ⁡ ( V ) be a unitary representation of Γ on a finite-dimensional complex vector space V and let v be a multiplier system on Γ of weight k. We then say that 𝒳 ρ , v = v ⋅ ρ is a matrix-valued multiplier system of weight k on Γ.

One of the most important multiplier systems is the so-called theta multiplier v θ which is associated with the Jacobi theta function

θ ⁢ ( τ ) = 1 + 2 ⁢ ∑ n = 1 ∞ e 2 ⁢ π ⁢ i ⁢ n 2 ⁢ τ .

More precisely,

θ ⁢ ( A ⁢ τ ) = j A ⁢ ( τ ) 1 / 2 ⁢ v θ ⁢ ( A ) ⁢ θ ⁢ ( τ ) for ⁢ A ∈ Γ 0 ⁢ ( 4 ) .

Definition 2.1.

The theta multiplier v θ is multiplier system of weight 1 2 defined on Γ 0 ⁢ ( 4 ) by

v θ ⁢ ( ( a b c d ) ) = ( c d ) ⁢ ε d - 1 ,

where ε d = 1 if d ≡ 1 ( mod 4 ) , and i else.

2.5 Cusps and stabilizers

Let N be a positive integer and let

C 0 ⁢ ( N ) = { 𝔞 1 , … , 𝔞 r } ≃ ℙ 1 ⁢ ( ℚ ) / Γ 0 ⁢ ( N )

be a fixed set of representatives for the cusps of Γ 0 ⁢ ( N ) chosen such that 𝔞 1 = ∞ = ( 1 : 0 ) and for 2 ≤ i ≤ r we set 𝔞 i = ( a i : c i ) ≃ a i / c i , where 0 < c i < N , c i ∣ N and a i ≡ a i ′ ( mod gcd ⁡ ( c i , N / c i ) ) with gcd ⁡ ( a i , c i ) = 1 and a i ′ runs through representatives of primitive residue classes modulo gcd ⁡ ( c i , N / c i ) . That this is indeed a set of representatives follows for instance from [5, Corollary 6.3.23]

For each cusp 𝔞 = ( a : c ) ∈ ℙ 1 ( ℚ ) , we choose a cusp normalizing map A 𝔞 = ( a b c d ) ∈ SL 2 ⁡ ( ℤ ) such that A 𝔞 ⁢ ( ∞ ) = 𝔞 and the stabilizer of 𝔞 in Γ 0 ⁢ ( N ) is infinite cyclic and generated by

T 𝔞 = A 𝔞 ⁢ T w 𝔞 ⁢ A 𝔞 - 1 = ( 1 - c ⁢ a ⁢ w 𝔞 a 2 ⁢ w 𝔞 - c 2 ⁢ w 𝔞 1 + a ⁢ c ⁢ w 𝔞 ) ,

where w 𝔞 = N / gcd ⁡ ( N , c 2 ) is the width of the cusp 𝔞 and T = ( 1 1 0 1 ) . Note that T 𝔞 ∈ Γ 0 ⁢ ( N ) ∩ Γ ⁢ ( w 𝔞 ) , and for 𝔞 = ( 1 : 0 ) = ∞ we choose A ∞ = 1 2 and T ∞ = T .

3 The Weil representation and modular forms

3.1 The Weil representation

Let D = ( D , Q ) be a finite quadratic module and let { 𝐞 γ } γ ∈ D be a basis of the group algebra

ℂ ⁢ [ D ] = { ∑ γ ∈ D c γ ⁢ 𝐞 γ | c γ ∈ ℂ } ,

which we usually just view as a vector space, namely ℂ ⁢ [ D ] ≃ ℂ n , where n = | D | and with Hermitian inner product defined by 〈 𝐞 γ , 𝐞 α 〉 = 1 if γ = α , and 0 otherwise.

The Weil representation associated with D, denoted by ρ D , is a unitary representation of Mp 2 ⁡ ( ℤ ) on the group algebra ℂ ⁢ [ D ] that can be defined by the action

ρ D ⁢ ( T ) ⁢ 𝐞 γ = e ⁢ ( Q ⁢ ( γ ) ) ⁢ 𝐞 γ ,

ρ D ⁢ ( S ) ⁢ 𝐞 γ = σ ⁢ ( D ) | D | ⁢ ∑ δ ∈ D e ⁢ ( - B ⁢ ( γ , δ ) ) ⁢ 𝐞 δ ,

where σ ⁢ ( D ) = e 8 ⁢ ( - sign ⁡ ( D ) ) . It is easy to verify by a direct computation that

ρ D ⁢ ( Z ) ⁢ 𝐞 γ = e 4 ⁢ ( - sign ⁡ ( D ) ) ⁢ 𝐞 - γ and ρ D ⁢ ( Z 2 ) ⁢ 𝐞 γ = e 2 ⁢ ( - sign ⁡ ( D ) ) ⁢ 𝐞 γ = ( - 1 ) sign ⁡ ( D ) ⁢ 𝐞 γ .

Since Z 2 = ( I 2 , - 1 ) , it is clear that ρ D factors through a representation of SL 2 ⁡ ( ℤ ) if and only if sign ⁡ ( D ) is even.

Using the Jordan decomposition of D, it is possible to find explicit formulas for the matrix coefficients of the Weil representation. For more details regarding the formulas below, see, e.g., [17].

Definition 3.2.

Let N be the level of D and define χ D : Γ → ℂ by

χ D ⁢ ( ( a b c d ) ) = ( | D | d ) if ⁢ 4 ∤ N ,

and otherwise

χ D ⁢ ( ( a b c d ) ) = ( | D | d ) ⋅ { 1 , sign ⁡ ( D ) ≡ 0 ( mod 4 ) , ( 2 d ) ⁢ v θ ⁢ ( A ) , sign ⁡ ( D ) ≡ 1 ( mod 4 ) , ( - 1 d ) , sign ⁡ ( D ) ≡ 2 ( mod 4 ) , ( 2 d ) ⁢ v ¯ θ ⁢ ( A ) , sign ⁡ ( D ) ≡ 3 ( mod 4 ) .

Note that if A ∈ Γ ⁢ ( N ) , then χ D ⁢ ( A ) = 1 if the signature is even, and ( c d ) else.

Lemma 3.3.

If A = ( a b c d ) ∈ Γ , then

ρ D ⁢ ( A ) ⁢ 𝐞 γ = χ D ⁢ ( A ) ⁢ e ⁢ ( b ⁢ d ⁢ Q ⁢ ( γ ) ) ⁢ 𝐞 a ⁢ γ if ⁢ c ≡ 0 ( mod N ) ,

and otherwise

ρ D ⁢ ( A ) ⁢ 𝐞 γ = ξ D ⁢ ( A ) ⁢ | D c | | D | ⁢ ∑ α ∈ D / D c f A ⁢ ( γ , α ) ⁢ e ⁢ ( B ⁢ ( x c , b ⁢ γ + α ) ) ⁢ 𝐞 d ⁢ γ + c ⁢ α + x c ,

where ξ D ⁢ ( A ) is an explicit 8-th root of unity, D c = { γ ∈ D ∣ c ⁢ γ = 0 } , x c ∈ D satisfies

c ⁢ Q ⁢ ( y ) = B ⁢ ( x c , y ) for all ⁢ y ∈ D c ,

and

f A ⁢ ( γ , α ) = e ⁢ ( b ⁢ d ⁢ Q ⁢ ( γ ) + a ⁢ c ⁢ Q ⁢ ( α ) + b ⁢ c ⁢ B ⁢ ( γ , α ) ) .

Remark 3.4.

The precise formula for the 8-th root of unity ξ D ⁢ ( A ) is not important here, other than for constructing examples. We therefore refer to [17] for all details and simply note that there is a typo in the formula for ξ 2 (in the notation of [17]) where c 2 should be - c 2 , and the case of c even and sign ⁡ ( D ) ≡ 2 ( mod 4 ) was overlooked. The corrected definition is to set ξ 2 = 1 if c is odd, and otherwise let

ξ 2 = e 8 ⁢ ( ( a + 1 ) ⁢ sign ⁡ ( D ⁢ ( 2 ) ) ) ⁢ { 1 if ⁢ sign ⁡ ( D ) ⁢ is even, e 8 ⁢ ( ( a + 1 ) ⁢ ( c 2 - 1 ) ) if ⁢ sign ⁡ ( D ) ⁢ is odd.

Finally, for the formula to be valid for both positive and negative values of c when the signature is odd, it is necessary to generalize [17, Lemma 4.4], leading to an additional factor of ( - 1 , c ) ∞ sign ⁡ ( D ) ( = - 1 if c < 0 and sign ⁡ ( D ) is odd, and otherwise 1).

To avoid the metaplectic group when the signature is odd, we will from now on let ρ D : Γ → GL ( ℂ [ D ] ) denote the corresponding matrix-valued multiplier system given by the section s. Note that ρ D ⁢ ( A ) = 𝒳 D ⋅ χ D , where 𝒳 D is a unitary representation of Γ and χ D is defined above.

4 Modular forms

4.1 Classical modular forms

Let G ⩽ Γ be a finite index subgroup, let k ∈ 1 2 ⁢ ℤ and let v : G → ℂ × be a multiplier system of weight k. The space of holomorphic modular forms of weight k and multiplier system v on G is defined as usual with M k ⁢ ( G , v ) consisting of holomorphic functions f : ℍ → ℂ that satisfy the following conditions:

f | k ⁢ A ⁢ ( τ ) ≔ j A ⁢ ( τ ) - k ⁢ f ⁢ ( A ⁢ τ ) = v ⁢ ( A ) ⁢ f ⁢ ( τ ) for all A ∈ G and τ ∈ ℍ .
f | k ⁢ B ⁢ ( τ ) is holomorphic in τ as ℑ ⁡ ( τ ) → ∞ for all B ∈ Γ .

Note that the second condition implies that f extends to a holomorphic function on ℍ ∪ ℙ 1 ⁢ ( ℚ ) . The subspace of cusp forms S k ⁢ ( G , v ) consists of those functions that also satisfy f | k ⁢ B ⁢ ( τ ) → 0 as ℑ ⁡ τ → ∞ for all B ∈ Γ . It is important to remember that the operator B ↦ | k B is in general only a group action of Mp 2 ⁡ ( ℤ ) and that for A , B ∈ Γ ,

f | k ⁢ A ⁢ B = σ k ⁢ ( A , B ) ⁢ f | k ⁢ A | k ⁢ B ,

where, of course, σ k ⁢ ( A , B ) = 1 if k is an integer.

4.2 Vector-valued modular forms for the Weil representation

Let ( D , Q ) be a finite quadratic module of level N and let ρ D be the associated Weil representation viewed as a matrix-valued multiplier system. The space of vector-valued modular forms for ρ D is denoted by M k ⁢ ( ρ D ) and consists of those functions f : ℍ → ℂ ⁢ [ D ] that can be written as f = ∑ γ ∈ D f γ ⁢ 𝐞 γ and satisfy the following conditions:

f | k ⁢ A = ρ D ⁢ ( A ) ⁢ f for all A ∈ Γ .
f γ ∈ M k ⁢ ( Γ ⁢ ( N ) , χ D ) for all γ ∈ D .

Other spaces, for instance of vector-valued cusp forms S k ⁢ ( ρ D ) , weakly holomorphic modular forms M k ! ⁢ ( ρ D ) , or harmonic weak Maass forms H k ⁢ ( ρ D ) , can be defined in the same way, by replacing M k ⁢ ( Γ ⁢ ( N ) , χ D ) by the appropriate space in (ii).

If ρ D is a matrix-valued multiplier system of weight k, it is possible to simplify the notation by defining a different ( k , ρ D ) slash-action on functions f : ℍ → V by setting

f | k , ρ D ⁢ A ⁢ ( τ ) = ρ D ⁢ ( A ) - 1 ⁢ f | k ⁢ A ⁢ ( τ ) = j A ⁢ ( τ ) - k ⁢ ρ D ⁢ ( A ) - 1 ⁢ f ⁢ ( A ⁢ τ )

and observing that this is now a true group action

f | k , ρ D ⁢ A | k , ρ D ⁢ B ⁢ ( τ ) = ρ D ⁢ ( A ⁢ B ) - 1 ⁢ f | k ⁢ A ⁢ B = f | k , ρ D ⁢ A ⁢ B ⁢ ( τ ) .

The central element Z must act trivially on any non-zero f ∈ M k ⁢ ( ρ D ) , and it is easy to see that f | k , ρ D ⁢ Z = f if and only if

e - π ⁢ i ⁢ k ⁢ f γ = e 4 ⁢ ( - sign ⁡ ( D ) ) ⁢ f - γ .

If ε = e 4 ⁢ ( 2 ⁢ k - sign ⁡ ( D ) ) , then f γ = ε ⁢ f - γ , and since ε 2 = 1 , it follows that 2 ⁢ k ≡ sign ⁡ ( D ) ( mod 2 ) . In other words, if we define

M k ε ⁢ ( ρ D ) = { f ∈ M k ⁢ ( ρ D ) ∣ f - γ = ε ⁢ f γ } ,

then M k ⁢ ( ρ D ) is only non-trivial if

(4.1) k ≡ 1 2 ⁢ sign ⁡ ( D ) ( mod 1 ) ,

and in this case

M k ⁢ ( ρ D ) = M k ε ⁢ ( ρ D ) with ⁢ ε = ( - 1 ) k - 1 2 ⁢ sign ⁡ ( D ) .

As has been hinted at before, it is clear from (4.1) that even and odd signatures correspond to integral and half-integral weight modular forms, respectively.

If f ∈ M k ⁢ ( ρ D ) has components f γ , we see that

f γ ⁢ ( τ + 1 ) = e ⁢ ( Q ⁢ ( γ ) ) ⁢ f γ ⁢ ( τ ) ,

and it follows that f γ has a Fourier expansion of the form

f γ ⁢ ( τ ) = ∑ n ∈ ℤ , n γ = n + Q ⁢ ( γ ) ≥ 0 c ⁢ ( γ , n ) ⁢ q n γ .

A group homomorphism ψ : D → D which also preserves the quadratic form, in other words, Q ⁢ ( ψ ⁢ ( α ) ) = Q ⁢ ( α ) for all α ∈ D , is said to be an automorphism of D and it induces an action on M k ⁢ ( ρ D ) by

ψ ⁢ ( ∑ γ ∈ D f γ ) = ∑ γ ∈ D f ψ ⁢ ( γ ) .

5 Main results and proofs

Let ( D , Q ) be a finite quadratic module of level N and let ρ D be the associated Weil representation. If γ ∈ D is an isotropic element, then Q ⁢ ( γ ) = 0 , and it follows from Lemma 3.3 that if A ∈ Γ 0 ⁢ ( N ) , then

ρ D ⁢ ( A ) ⁢ 𝐞 γ = χ D ⁢ ( A ) ⁢ e ⁢ ( b ⁢ d ⁢ Q ⁢ ( γ ) ) ⁢ 𝐞 a ⁢ γ = χ D ⁢ ( A ) ⁢ 𝐞 a ⁢ γ .

It follows immediately that if

f = ∑ γ ∈ D f γ ⁢ 𝐞 γ ∈ M k ⁢ ( ρ D ) ,

then f 0 ∈ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) and f γ ∈ M k ⁢ ( Γ 1 ⁢ ( N ) , v ) for all γ ∈ D , where v is a multiplier system defined by

v ⁢ ( a b c d ) = e ⁢ ( b ⁢ Q ⁢ ( γ ) ) ⋅ { ( c d ) if ⁢ sign ⁡ ( D ) ⁢ is odd, 1 if ⁢ sign ⁡ ( D ) ⁢ is even.

Note that e ⁢ ( Q ⁢ ( γ ) ) is an N-th root of unity and the map

( a b c d ) ↦ e ⁢ ( b ⁢ Q ⁢ ( γ ) )

is a character on Γ 1 ⁢ ( N ) induced by a non-Dirichlet character on Γ 1 ⁢ ( N ) / Γ ⁢ ( N ) ≃ ℤ / N ⁢ ℤ .

The result above for f 0 is a special case of the following more general construction.

Proposition 5.5.

Let ( D , Q ) be a finite quadratic module of level N with associated Weil representation ρ D and let

f = ∑ γ ∈ D f γ ⁢ 𝐞 γ ∈ M k ⁢ ( ρ D ) .

If S 0 ⩽ D is an isotropic subgroup and we set

Φ S 0 ⁢ ( f ) = ∑ β ∈ S 0 f β ,

then Φ S 0 ⁢ ( f ) ∈ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) .

Proof.

If A = ( a b c d ) ∈ Γ 0 ⁢ ( N ) and β ∈ S 0 , then

f β | k ⁢ A = j A ⁢ ( τ ) - k ⁢ f β ⁢ ( A ⁢ τ ) = χ D ⁢ ( A ) ⁢ e ⁢ ( b ⁢ d ⁢ Q ⁢ ( a - 1 ⁢ β ) ) ⁢ f a - 1 ⁢ β = χ D ⁢ ( A ) ⁢ f a - 1 ⁢ β ,

where a - 1 is the inverse of a modulo the order of D. Since a is also coprime to the order of S 0 , it follows that

Φ S 0 ⁢ ( f ) | k ⁢ A = ∑ β ∈ S 0 f β | k ⁢ A = χ D ⁢ ( A ) ⁢ ∑ β ∈ S 0 f a - 1 ⁢ β = χ D ⁢ ( A ) ⁢ F S 0 .

This concludes the proof. ∎

The converse lift of a scalar to a vector-valued modular form is more technical.

Theorem 5.6.

Let ( D , Q ) be a finite quadratic module of level N, let F ∈ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) and let S 0 ⩽ D be an isotropic subgroup. Define

Ψ S 0 ⁢ ( F ) ⁢ ( τ ) = ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ 𝐜 M ⁢ F | k ⁢ M ⁢ ( τ ) , where ⁢ 𝐜 M = ρ D ⁢ ( M ) - 1 ⁢ ∑ γ ∈ S 0 𝐞 γ .

Then Ψ S 0 ⁢ ( F ) ∈ M k ⁢ ( ρ D ) and if F is a cusp form, then so is Ψ S 0 ⁢ ( F ) . Furthermore, Ψ S 0 ⁢ ( F ) is invariant under all automorphisms of D which stabilize S 0 .

Proof.

Let A , B ∈ SL 2 ⁡ ( ℤ ) and set 𝐒 0 = ∑ β ∈ S 0 𝐞 β . Then 𝐜 A = ρ D ⁢ ( A ) - 1 ⁢ 𝐒 0 and

𝐜 A ⁢ B = ρ D ⁢ ( A ⁢ B ) - 1 ⁢ 𝐒 0

= σ k ⁢ ( A , B ) ⁢ ( ρ D ⁢ ( A ) ⁢ ρ D ⁢ ( B ) ) - 1 ⁢ 𝐒 0

= σ k ⁢ ( A , B ) ⁢ ρ D ⁢ ( B ) - 1 ⁢ ρ D ⁢ ( A ) - 1 ⁢ 𝐒 0

= σ k ⁢ ( A , B ) ⁢ ρ D ⁢ ( B ) - 1 ⁢ 𝐜 A

= σ k ⁢ ( A , B ) ⁢ σ k ⁢ ( B , B - 1 ) ⁢ ρ D ⁢ ( B - 1 ) ⁢ 𝐜 A .

Let C ∈ Γ 0 ⁢ ( N ) . It is easy to see that

𝐜 C = ρ D ⁢ ( C ) - 1 ⁢ 𝐒 0 = χ D ⁢ ( C ) ¯ ⁢ 𝐒 0 ,

and therefore

𝐜 C ⁢ B = ρ D ⁢ ( C ⁢ B ) - 1 ⁢ 𝐒 0 = σ k ⁢ ( C , B ) ⁢ ( ρ D ⁢ ( C ) ⁢ ρ D ⁢ ( B ) ) - 1 ⁢ 𝐒 0 = χ D ⁢ ( A ) ¯ ⁢ σ ⁢ ( C , B ) ⁢ 𝐜 B .

It follows that 𝐜 C ⁢ B ⁢ F | k ⁢ C ⁢ B = 𝐜 B ⁢ F | k ⁢ B , in other words, the function Ψ S 0 ⁢ ( F ) is well-defined and independent of the choice of coset-representatives for Γ 0 ⁢ ( N ) ∖ Γ . To verify that Ψ S 0 ⁢ ( F ) ∈ M k ⁢ ( ρ D ) , we first note that

Ψ S 0 ⁢ ( F ) | k ⁢ B = ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ 𝐜 M ⁢ F | k ⁢ M | k ⁢ B ⁢ ( τ )

= ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ 𝐜 M ⁢ σ k ⁢ ( M , B ) - 1 ⁢ F | k ⁢ M ⁢ B ⁢ ( τ )

= ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ σ k ⁢ ( M ⁢ B - 1 , B ) - 1 ⁢ 𝐜 M ⁢ B - 1 ⁢ F | k ⁢ M ⁢ ( τ )

= ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ σ k ⁢ ( M ⁢ B - 1 , B ) - 1 ⁢ σ k ⁢ ( M , B - 1 ) ⁢ σ k ⁢ ( B - 1 , B ) ⁢ ρ D ⁢ ( B ) ⁢ 𝐜 M ⁢ F | k ⁢ M ⁢ ( τ )

= ρ D ⁢ ( B ) ⁢ ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ 𝐜 M ⁢ F | k ⁢ M ⁢ ( τ )

= ρ D ⁢ ( B ) ⁢ Ψ S 0 ⁢ ( F ) .

Furthermore, if we write Ψ S 0 ⁢ ( F ) = ∑ γ ∈ D f γ , then

f γ = ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ c M , γ ⁢ F | k ⁢ M ,

where

c M , γ = ∑ β ∈ 𝐒 0 ρ D ⁢ ( M ) γ ⁢ β - 1

and any regularity conditions of F at the cusps of Γ are also satisfied by f γ . The conclusion of the theorem follows by the final observation that 𝐒 0 is invariant under any automorphism which stabilizes S 0 . ∎

Remark 5.7.

Note that the above theorem holds also when S 0 ⊂ D is an isotropic subset which is invariant under multiplication by ( ℤ / N ⁢ ℤ ) * . This is the version which was first given in [12].

Theorem 5.8.

Let ( D , Q ) be a finite quadratic module of level N, let F ∈ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) and let S 0 ⩽ D be an isotropic subgroup. Assume that the Fourier expansion of F at the cusp a i is ∑ m ∈ Z + α i b i ⁢ ( m ) ⁢ q m / w i , where w i is the cusp width and α ∈ [ 0 , 1 [ . If we write Ψ S 0 ⁢ ( F ) = ∑ α ∈ D f α ⁢ e α , then

f α ⁢ ( τ ) = ∑ n ∈ ℤ + Q ⁢ ( α ) c ⁢ ( α , n ) ⁢ q n ,

with

c ⁢ ( α , n ) = ∑ i = 1 r Λ i ⁢ ( α ) ⁢ b i ⁢ ( n ⁢ w i ) , n ∈ ℤ + Q ⁢ ( α ) ,

where the sum is taken over C 0 ⁢ ( N ) and the complete set of cusp representatives for Γ 0 ⁢ ( N ) and the constants Λ i ⁢ ( α ) are explicitly given by

Λ i ⁢ ( α ) = | D c i | | D | ⁢ ξ D ⁢ ( A i - 1 ) ⁢ w i ⁢ ∑ ω ∈ D c i ⁣ * ∩ D S 0 c i ⁣ * ∩ { α + S 0 } Φ S 0 , a i , c i ⁢ ( ω ) ⁢ e ⁢ ( - d i ⁢ Q c i ⁢ ( ω ) ) ,

Φ S 0 , a , c ⁢ ( ω ) = ∑ γ ∈ c - 1 ⁢ ( S 0 ) / D c e ⁢ ( - a ⁢ c ⁢ Q ⁢ ( γ ) + B ⁢ ( ω , γ ) ) ,

where A i = ( a i b i c i d i ) is a fixed cusp normalizing map for the cusp a i ,

Remark 5.9.

In the case when S 0 = { 0 } , then Λ i ⁢ ( α ) = 0 if α ∉ D c i ⁣ * , and otherwise

Λ i ⁢ ( α ) = | D c i | | D | ⁢ ξ D ⁢ ( A i - 1 ) ⁢ w i ⁢ e ⁢ ( - d i ⁢ Q c i ⁢ ( α ) ) .

Proof of Theorem 5.8.

It follows from the construction that if we write Ψ S 0 ⁢ ( F ) = ∑ α ∈ D f α ⁢ 𝐞 α , then

f α = ∑ M ∈ Γ 0 ⁢ ( N ) ∖ Γ 〈 𝐜 M , 𝐞 α 〉 ⁢ F | k ⁢ M ,

and we will consider separately the explicit evaluation of 〈 𝐜 M , 𝐞 α 〉 and F | k ⁢ M . Choose the same set of cusp representatives as before, that is, C 0 ⁢ ( N ) = { 𝔞 1 , … , 𝔞 r } , and to each cusp 𝔞 i there is an associated cusp normalizing map σ i , a cusp width w i = w 𝔞 i , and a generator of the stabilizer T 𝔞 i = σ i ⁢ T w i ⁢ σ i - 1 . It is well known that the set of matrices

{ M i ⁢ j = A i ⁢ T j ∣ 1 ≤ i ≤ r , 0 ≤ j ≤ w 𝔞 i - 1 }

form a complete set of representatives for Γ 0 ⁢ ( N ) ∖ Γ . Hence,

f α = ∑ M i ⁢ j 〈 𝐜 M i ⁢ j , 𝐞 α 〉 ⁢ F | k ⁢ M i ⁢ j = ∑ i = 0 r ∑ j = 0 w i - 1 〈 𝐜 A i ⁢ T j , 𝐞 α 〉 ⁢ F | k ⁢ A i ⁢ T j .

Define α i ∈ [ 0 , 1 [ by χ D ⁢ ( T i ) = e ⁢ ( α i ) . Then F has the following Fourier expansion at the cusp 𝔞 i :

F i ⁢ ( τ ) = F | k ⁢ A i ⁢ ( τ ) = ∑ n ∈ ℤ + α i b i ⁢ ( n ) ⁢ q n w i .

Further, we can collect terms with the same fractional q-powers by writing

F i ⁢ ( τ ) = ∑ t ⁢ ( mod ⁢ w i ) ∑ n ≡ t + α i ⁢ ( mod ⁢ w i ) b i ⁢ ( n ) ⁢ q n w i = ∑ t ⁢ ( mod ⁢ w i ) ∑ n ∈ ℤ + θ i , t b i ⁢ ( n ⁢ w i ) ⁢ q n = ∑ t ⁢ ( mod ⁢ w i ) F i , t ⁢ ( τ ) ,

where θ i , t = ( t + α i ) / w i . Clearly, if j ∈ ℤ , then F i , t ⁢ ( τ + j ) = e ⁢ ( j ⁢ θ i , t ) ⁢ F i , t ⁢ ( τ ) and

f α ⁢ ( τ ) = ∑ i = 0 r ∑ j = 0 w i - 1 〈 𝐜 M i ⁢ j , 𝐞 α 〉 ⁢ ∑ t ⁢ ( mod ⁢ w i ) F i , t ⁢ ( τ + j ) = ∑ i = 0 r ∑ j = 0 w i - 1 〈 𝐜 M i ⁢ j , 𝐞 α 〉 ⁢ ∑ t ⁢ ( mod ⁢ w i ) e ⁢ ( j ⁢ θ i , t ) ⁢ F i , t ⁢ ( τ ) .

The proof is completed by applying Lemma 5.17. ∎

5.1 Cusp stabilizers and character values

To complete the proof of Theorem 5.8, we will need to explicitly evaluate χ D ⁢ ( T 𝔞 ) for any cusp representative 𝔞 = ( a : c ) and compare this with the values of e ⁢ ( w 𝔞 ⁢ Q ⁢ ( γ ) ) for γ ∈ D c ⁣ * . In this way, we will be able to collate the coefficients of F at different cups with coefficients of f at different components.

To evaluate χ D ⁢ ( T 𝔞 ) , we follow a similar approach as in [13] except that we consider the indecomposable Jordan components individually. There are also a few additional cases to consider since we include the case of odd signature. The first step is to observe that χ D is multiplicative with respect to a Jordan decomposition of D and it is therefore sufficient to study each type of component separately.

Lemma 5.10.

If p > 2 is a prime dividing N and k and t are positive integers, then

χ A p k t ⁢ ( T 𝔞 ) = 1

for all cusps a of Γ 0 ⁢ ( N ) .

Proof.

Since sign ⁡ ( A p k t ) ≡ 1 - p k ( mod 4 ) , if 𝔞 = ( a : c ) , then p ∣ a w 𝔞 and

χ A p k t ⁢ ( T 𝔞 ) = ( p k 1 + a ⁢ c ⁢ w 𝔞 ) × { 1 , p k ≡ 1 ( mod 4 ) - 1 , p k ≡ 3 ( mod 4 ) } = 1 ,

where we used that if n is odd, then

( p 1 + p ⁢ n ) = 1 if ⁢ p ≡ 1 ( mod 4 ) ,

and -1 else. ∎

Lemma 5.11.

If k is a positive integer, then

χ B 2 k ⁢ ( T 𝔞 ) = χ C 2 k ⁢ ( T 𝔞 ) = 1

for all cusps a of Γ 0 ⁢ ( N ) .

Proof.

Since

sign ⁡ ( B 2 k ) = sign ⁡ ( C 2 k ) ≡ 0 ( mod 4 )

and the order of each of these components is 2 2 ⁢ k , it follows that

χ B 2 k ⁢ ( T 𝔞 ) = χ C 2 k ⁢ ( T 𝔞 ) = ( 2 2 ⁢ k 1 + a ⁢ c ⁢ w 𝔞 ) = ( 2 1 + a ⁢ c ⁢ w 𝔞 ) 2 ⁢ k = 1 .

This concludes the proof. ∎

Lemma 5.12.

If k is a positive integer and t is odd, then χ A 2 k t ⁢ ( T a ) = 1 unless one of the following cases occurs:

2 ∥ c and 8 ∥ N , in which case χ A 2 k t ⁢ ( T 𝔞 ) = ( - 1 ) k .
4 ∥ c , 4 ∣ N and 32 ∤ N , in which case χ A 2 k t ⁢ ( T 𝔞 ) = ( - 1 ) k + 1 .
2 ∥ c and 4 ∥ N , in which case we have χ A 2 k t ⁢ ( T 𝔞 ) = i w 𝔞 ⁢ t .

Proof.

First observe that if there is a non-trivial component of the type A 2 k t , then 4 ∣ N and for a cusp 𝔞 = ( a : c ) we can write

χ A 2 k t ⁢ ( T 𝔞 ) = ( 2 k + 1 1 + a ⁢ c ⁢ w ( 𝔞 ) ⁢ ( - c 2 ⁢ w 𝔞 1 + a ⁢ c ⁢ w 𝔞 ) ⋅ { ε 1 + a ⁢ c ⁢ w 𝔞 - 1 , t ≡ 1 ( mod 4 ) , ε 1 + a ⁢ c ⁢ w 𝔞 , t ≡ 3 ( mod 4 ) ,

= ( 2 1 + a ⁢ c ⁢ w 𝔞 ) k + 1 + v 2 ⁢ ( w 𝔞 ) ⁢ ∏ 2 ⁢ < p ∣ ⁢ w 𝔞 ( p 1 + a ⁢ c ⁢ w 𝔞 ) v p ⁢ ( w 𝔞 ) ⁢ ( - 1 1 + a ⁢ c ⁢ w 𝔞 ) 1 - ( - 1 t ) ⁢ 1 / 2 .

Recall that gcd ⁡ ( a , c ) = 1 . The individual factors can now be evaluated explicitly using known properties of the Kronecker symbols:

( 2 1 + a ⁢ c ⁢ w 𝔞 ) = { 1 , 8 ∣ c w 𝔞 , - 1 , 4 ∥ c w 𝔞 , - ( - 1 a ⁢ c ⁢ w 𝔞 / 2 ) , 2 ∥ c w 𝔞 ,

( p 1 + a ⁢ c ⁢ w 𝔞 ) = { 1 , p ≡ 1 ( mod 4 ) , 1 , p ≡ 3 ( mod 4 ) and 4 ∣ c w 𝔞 , - 1 , p ≡ 3 ( mod 4 ) and 2 ∥ c w 𝔞 ,

( - 1 1 + a ⁢ c ⁢ w 𝔞 ) = { 1 , 4 ∣ c w 𝔞 , - 1 , 2 ∥ c w 𝔞 .

It is clear that

χ A 2 k t ( T 𝔞 ) = 1 if 8 ∣ c w 𝔞 ,

and there are only three other possibilities:

If 2 ∥ c and 4 ∥ N , then 2 ∥ c w 𝔞 , 2 ∤ w 𝔞 , k = 1 , v 2 ⁢ ( w 𝔞 ) = 0 , and therefore

χ A 2 k t ⁢ ( T 𝔞 ) = ∏ p ≡ 3 ( mod 4 ) , p ∣ w 𝔞 ( - 1 ) v p ⁢ ( w 𝔞 ) ⁢ ( - 1 ) 1 - ( - 1 t ) ⁢ 1 / 2 = ( - 1 w 𝔞 ) ⁢ i t = i t ⁢ w 𝔞 .
If 2 ∥ c and 8 ∥ N , then 4 ∥ c w 𝔞 , 2 ∥ w 𝔞 , 1 ≤ k ≤ 2 , v 2 ⁢ ( w 𝔞 ) = 1 , and therefore

χ A 2 k t ⁢ ( T 𝔞 ) = ( - 1 ) k .
If 4 ∥ c , 4 ∣ N and 32 ∤ N , then 4 ∥ c w 𝔞 , 2 ∤ w 𝔞 , 1 ≤ k ≤ 3 , v 2 ⁢ ( w 𝔞 ) = 0 , and therefore

χ A 2 k t ⁢ ( T 𝔞 ) = ( - 1 ) k + 1 .

This concludes the proof. ∎

Combining the three previous lemmas and using the multiplicativity of χ D and additivity of the signatures, we conclude the following proposition.

Proposition 5.13.

If a = ( a : c ) is a cusp of Γ 0 ⁢ ( N ) and T a is as above, then χ D ⁢ ( T a ) = 1 unless the Jordan decomposition of D has at least one component of the form A 2 k t with k = v 2 ⁢ ( c ) ≥ 1 and either one of the following cases occurs:

2 ∥ c and 8 ∥ N or 4 ∥ c , 4 ∣ N and 32 ∤ N , and sign ⁡ ( D ⁢ [ 2 k ] ) is odd, in which case χ D ⁢ ( T 𝔞 ) = - 1 .
2 ∥ c and 4 ∥ N and sign ⁡ ( D ⁢ [ 2 k ] ) ⁢ w p ≢ 0 ( mod 4 ) , in which case

χ D ⁢ ( T 𝔞 ) = i sign ⁡ ( D ⁢ [ 2 k ] ) ⁢ w p .

Proof.

To conclude the proof, we consider the three cases from Lemma 5.6:

If 2 ∥ c and 8 ∥ N , then 1 ≤ k ≤ 2 and

χ A 2 k t ⁢ ( T 𝔞 ) = ( - 1 ) k = - 1

only if k = 1 .
If 4 ∥ c , 4 ∣ N and 32 ∤ N , then 1 ≤ k ≤ 3 and

χ A 2 k t ⁢ ( T 𝔞 ) = ( - 1 ) k + 1 = - 1

only if k = 2 .
If 2 ∥ c and 4 ∥ N , then k = 1 and

χ A 2 t ⁢ ( T 𝔞 ) = i w 𝔞 ⁢ t ,

which is ≠ 1 only if w 𝔞 ⁢ t ≢ 0 ( mod 4 ) .

This concludes the proof. ∎

The result now follows by considering the direct sum of all components of order 2 k .

Remark 5.14.

It is easy to verify that the above result agrees with that of [13, Proposition 5.1] in the case when the signature of D is even.

Proposition 5.15.

Let ( D , Q ) be a finite quadratic module of level N and let a = ( a : c ) be a cusp representative for Γ 0 ⁢ ( N ) . Then

e ⁢ ( w 𝔞 ⁢ Q ⁢ ( γ ) ) = χ D ⁢ ( T 𝔞 ) for all ⁢ γ ∈ D c ⁣ * .

Proof.

Choose a Jordan decomposition of D and let x c be the canonical representative of D c ⁣ * / D c . Then Q ⁢ ( x c ) = 0 unless v 2 ⁢ ( c ) = k ≥ 0 and D has non-trivial Jordan components of order 2 k of the form

D ⁢ [ 2 k ] = A 2 k t 1 ⊕ ⋯ ⊕ A 2 k t n ,

in which case

Q ⁢ ( x c ) ≡ t ⁢ 2 k 8 ( mod 1 ) ,

where t ≡ t 1 + ⋯ + t n ( mod 8 ) . For details, see [17, 13]. Since w 𝔞 = N / gcd ⁡ ( N , c 2 ) , it is clear that

c 2 ⁢ w 𝔞 ∈ N ⁢ ℤ ,

and therefore c 2 ⁢ w 𝔞 ⁢ Q ⁢ ( γ ) ∈ ℤ for any γ ∈ D . It follows that for any γ = x c + c ⁢ δ ∈ D c ⁣ * we have w 𝔞 ⁢ Q ⁢ ( γ ) = w 𝔞 ⁢ Q ⁢ ( x c ) , and if 8 ∣ c w 𝔞 , then w 𝔞 ⁢ Q ⁢ ( x c ) = 0 ( mod 1 ) . The result then follows by studying the remaining cases in the same way as in the proof of Proposition 5.13. ∎

5.2 Completion of the proof of Theorem 5.8

Lemma 5.16.

Let A = ( a b c d ) ∈ SL 2 ⁡ ( Z ) with c > 0 , or c = 0 and d = 1 . Then

𝐜 A ⁢ T j = Ξ ⁢ ( A ) ⁢ ∑ ω ∈ D c ⁣ * ∩ D S 0 c ⁣ * e ⁢ ( - d ⁢ Q c ⁢ ( ω ) ) ⁢ ∑ ν ∈ S 0 / ( S 0 ∩ D c ) e ⁢ ( - j ⁢ Q ⁢ ( ω ) ) ⁢ Φ S 0 , a , c ⁢ ( ω ) ⁢ 𝐞 ν + ω ,

where

Φ S 0 , a , c ⁢ ( ω ) = ∑ γ ∈ c - 1 ⁢ ( S 0 ) / D c e ⁢ ( - a ⁢ c ⁢ Q ⁢ ( γ ) + B ⁢ ( ω , γ ) )

and

Ξ ⁢ ( A ) = ξ ⁢ ( A - 1 ) ⁢ | D c | | D | .

Proof.

Since c > 0 , or c = 0 and d = 1 , it can be shown that σ ⁢ ( A , T j ) = 1 , and hence

𝐜 A ⁢ T j = ρ D ⁢ ( T - j ) ⁢ 𝐜 A .

Further, since D - c = D c , D - c ⁣ * = D c ⁣ * and Q - c ⁢ ( β ) = - Q c ⁢ ( β ) , we see that

𝐜 A = σ ⁢ ( A , A - 1 ) ⁢ ρ D ⁢ ( A - 1 ) ⁢ ∑ γ ∈ S 0 𝐞 γ

= ρ D ⁢ ( d - b - c a ) ⁢ ∑ γ ∈ S 0 𝐞 γ

= ξ ⁢ ( A - 1 ) ⁢ | D - c | | D | ⁢ ∑ γ ∈ S 0 ∑ α ∈ D - c e ⁢ ( d ⁢ Q - c ⁢ ( α ) - b ⁢ d ⁢ Q ⁢ ( γ ) - b ⁢ B ⁢ ( γ , α ) ) ⁢ 𝐞 α + a ⁢ γ

= Ξ ⁢ ( A ) ⁢ ∑ γ ∈ S 0 ∑ α ∈ D c e ⁢ ( - d ⁢ Q c ⁢ ( α ) - b ⁢ d ⁢ Q ⁢ ( γ ) - b ⁢ B ⁢ ( γ , α ) ) ⁢ 𝐞 α + a ⁢ γ

= Ξ ⁢ ( A ) ⁢ ∑ γ ∈ S 0 ∑ α ∈ D c ⁣ * e ⁢ ( - d ⁢ Q c ⁢ ( α ) - b ⁢ B ⁢ ( γ , α ) ) ⁢ 𝐞 α + a ⁢ γ

= Ξ ⁢ ( A ) ⁢ ∑ α ∈ D c ⁣ * e ⁢ ( - d ⁢ Q c ⁢ ( α ) ) ⁢ ∑ γ ∈ S 0 e ⁢ ( - b ⁢ B ⁢ ( γ , α ) ) ⁢ 𝐞 α + a ⁢ γ .

To conclude the proof, we note that all arguments in the proofs of [13, Propositions 5.5 and 5.6 and Theorem 5.7] are immediately applicable from this point on, as the arguments do not depend on the signature or existence of non-trivial cocycles. Note that we do not use the decomposition for the sum Φ S 0 , a , c and are therefore missing the factor | S 0 ∩ c ⁢ S 0 ⊥ | in the resulting formula. ∎

Lemma 5.17.

Let a = ( a : c ) and define θ t = ( t + α ) / w a , where χ D ⁢ ( T a ) = e ⁢ ( α ) . Assume that G is a function on the upper half-plane such that

G ⁢ ( τ ) = ∑ n ∈ ℤ + α b ⁢ ( n ) ⁢ q n / w 𝑎𝑛𝑑 G t ⁢ ( τ ) = ∑ n ∈ ℤ + θ t b ⁢ ( n ⁢ w 𝔞 ) ⁢ q n t ( mod w 𝔞 ) ,

for some complex coefficients b ⁢ ( n ) . Then there exists t δ ( mod w a ) such that

∑ j = 0 w 𝔞 - 1 〈 𝐜 A 𝔞 ⁢ T j , 𝐞 δ 〉 ⁢ 1 w 𝔞 ⁢ ∑ t ⁢ ( mod ⁢ w 𝔞 ) e ⁢ ( j ⁢ θ t ) ⁢ G t ⁢ ( τ ) = G t δ ⁢ ( τ ) ⁢ Λ ⁢ ( δ , A 𝔞 ) ,

where

Λ ⁢ ( δ , A 𝔞 ) = Ξ ⁢ ( A 𝔞 ) ⁢ ∑ ω ∈ D c ⁣ * ∩ D S 0 c ⁣ * ∩ { δ + S 0 } Φ S 0 , a , c ⁢ ( ω ) ⁢ e ⁢ ( - d ⁢ Q c ⁢ ( ω ) )

with Ξ ⁢ ( A a ) and Φ S 0 , a , c ⁢ ( ω ) defined as in Lemma 5.16.

Proof.

It follows from Proposition 5.15 that we can define t ω ( mod w 𝔞 ) such that θ t ω = Q ⁢ ( ω ) for all ω ∈ D c ⁣ * . Then we observe that

1 w 𝔞 ⁢ ∑ j = 0 w 𝔞 - 1 e ⁢ ( j ⁢ ( θ t - Q ⁢ ( ω ) ) ) = { 1 if ⁢ t = t ω , 0 else.

Using this orthogonality relation together with the formula for 〈 𝐜 A 𝔞 ⁢ T j , 𝐞 δ 〉 from Lemma 5.16, we see that

1 w 𝔞 ⁢ ∑ j = 0 w 𝔞 - 1 〈 𝐜 A 𝔞 ⁢ T j , 𝐞 δ 〉 ⁢ ∑ t ⁢ ( mod ⁢ w 𝔞 ) e ⁢ ( j ⁢ θ t ) ⁢ G t ⁢ ( τ )

= Ξ ⁢ ( A 𝔞 ) ⁢ ∑ ω ∈ D c ⁣ * ∩ D S 0 c ⁣ * Φ S 0 , a , c ⁢ ( ω ) ⁢ e ⁢ ( - d ⁢ Q c ⁢ ( ω ) ) ⁢ G t ω ⁢ ( τ ) ⁢ ∑ ν ∈ S 0 / ( S 0 ∩ D c ) 〈 𝐞 ν + ω , 𝐞 δ 〉

= Ξ ⁢ ( A 𝔞 ) ⁢ ∑ ω ∈ D c ⁣ * ∩ D S 0 c ⁣ * ∩ { δ + S 0 } Φ S 0 , a , c ⁢ ( ω ) ⁢ e ⁢ ( - d ⁢ Q c ⁢ ( ω ) ) ⁢ G t ω ⁢ ( τ ) .

Since D S 0 c ⁣ * is orthogonal to S 0 , we have Q ⁢ ( ω ) = Q ⁢ ( δ ) for ω ∈ D S 0 c ⁣ * ∩ { δ + S 0 } , and thus G t ω ⁢ ( τ ) = G t δ ⁢ ( τ ) , which concludes the proof. ∎

6 Injectivity of the lifting maps

With notation as in Theorem 5.8, let F ∈ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) , let S 0 ⩽ D be an isotropic subgroup and consider the composition

Φ S 0 ∘ Ψ S 0 ⁢ ( F ) = ∑ α ∈ S 0 ( ∑ n ∈ ℤ + Q ⁢ ( α ) c ⁢ ( α , n ) ⁢ q n ) = ∑ 0 ≤ n ∈ ℤ q n ⁢ ∑ α ∈ S 0 c ⁢ ( α , n ) = ∑ 0 ≤ n ∈ ℤ a ⁢ ( n ) ⁢ q n ,

where a ⁢ ( n ) is given by

a ⁢ ( n ) = ∑ α ∈ S 0 ∑ i = 1 r Λ i ⁢ ( α ) ⁢ b i ⁢ ( n ⁢ w i ) .

Note that if α ∈ S 0 , then α + S 0 = S 0 and we therefore have ω ∈ D c i ⁣ * ∩ D S 0 c i ⁣ * ∩ S 0 in the sum for Λ i ⁢ ( α ) . If ω ∈ D S 0 c * ∩ S 0 for some integer c, then

c ⁢ Q ⁢ ( γ ) + B ⁢ ( ω , γ ) = c ⁢ Q ⁢ ( γ ) = 0 for all ⁢ γ ∈ c - 1 ⁢ S 0 ∩ S 0 ⊥ ,

and the intersection is therefore empty if there exists γ ∈ c - 1 ⁢ S 0 ∩ S 0 ⊥ such that c ⁢ Q ⁢ ( γ ) ≠ 0 , and otherwise D S 0 c * ∩ S 0 = S 0 . Now, if ω ∈ S 0 ∩ D c ⁣ * and c ⁢ Q ⁢ ( γ ) = 0 for all γ ∈ c - 1 ⁢ S 0 , then

Φ S 0 , a , c ⁢ ( ω ) = ∑ γ ∈ c - 1 ⁢ ( S 0 ) / D c e ⁢ ( - a ⁢ c ⁢ Q ⁢ ( γ ) + B ⁢ ( ω , γ ) ) = | c - 1 ⁢ S 0 / D c | ,

and if ω ∈ S 0 ∩ D c ⁣ * , then

Q c ⁢ ( ω ) = Q ⁢ ( ω ) - Q ⁢ ( x c ) = - Q ⁢ ( x c ) .

Assume now that c i ≠ 0 satisfies c i ⁢ Q ⁢ ( γ ) = 0 for all γ ∈ c i - 1 ⁢ S 0 ∩ S 0 ⊥ . Then

Λ i ⁢ ( α ) = | D c i | | D | ⁢ ξ D ⁢ ( A i - 1 ) ⁢ w i ⁢ ∑ ω ∈ D c i ⁣ * ∩ S 0 Φ S 0 , a i , c i ⁢ ( ω ) ⁢ e ⁢ ( - d i ⁢ Q c i ⁢ ( ω ) )

= | D c i | | D | ⁢ ξ D ⁢ ( A i - 1 ) ⁢ w i ⁢ | c i - 1 ⁢ S 0 / D c i | ⁢ | D c i * ∩ S 0 | ⁢ e ⁢ ( d i ⁢ Q ⁢ ( x c i ) )

= ξ D ⁢ ( A i - 1 ) ⁢ e 8 ⁢ ( d i ⁢ 2 v 2 ⁢ ( c i ) ⁢ sign ⁡ ( D ⁢ [ 2 v 2 ⁢ ( c i ) ] ) ) ⁢ | D c i | | D | ⁢ w i ⁢ | c i - 1 ⁢ S 0 / D c i | ⁢ | D c i * ∩ S 0 |

= ξ D ⁢ ( A i - 1 ) ⁢ e 8 ⁢ ( d i ⁢ 2 v 2 ⁢ ( c i ) ⁢ sign ⁡ ( D ⁢ [ 2 v 2 ⁢ ( c i ) ] ) ) ⁢ w i .

It follows that

a ⁢ ( n ) = | S 0 | ⁢ c 1 ⁢ ( n ) + | S 0 | ⁢ ∑ i ∈ I S 0 ∖ { 1 } ξ D ⁢ ( A i - 1 ) ⁢ e 8 ⁢ ( d i ⁢ 2 v 2 ⁢ ( c i ) ⁢ sign ⁡ ( D ⁢ [ 2 v 2 ⁢ ( c i ) ] ) ) ⁢ w i ⁢ b i ⁢ ( n ⁢ w i ) ,

where

I S 0 = { 1 ≤ i ≤ r ∣ D S 0 c i * ∩ S 0 ≠ ∅ } .

We arrive at the following result, which is difficult to make explicit in general, but can either be studied computationally or be given in specific examples.

Definition 6.18.

Define the subspace

A k , S 0 ε 2 , … , ε r ⁢ ( Γ 0 ⁢ ( N ) , χ D ) ⊆ M k ⁢ ( Γ 0 ⁢ ( N ) , χ D )

consisting of those functions F that have Fourier expansions F i = ∑ b i ⁢ ( n ) ⁢ q n / w i at cusp 𝔞 i = ( a i : c i ) with coefficients that satisfy

b i ⁢ ( n ⁢ w i ) = ε i ⁢ b 1 ⁢ ( n ) , i ∈ I S 0 ,

and ∑ i ∈ I S 0 ε i ≠ - 1 , where

I S 0 = { 1 ≤ i ≤ r ∣ D S 0 c i * ∩ S 0 ≠ ∅ }

and

ε i = ξ ¯ D ⁢ ( A i - 1 ) ⁢ e 8 ⁢ ( - d i ⁢ 2 v 2 ⁢ ( c i ) ⁢ sign ⁡ ( D ⁢ [ 2 v 2 ⁢ ( c i ) ] ) ) ⁢ w i - 1 ,

with notation as above.

Theorem 6.19.

The lifting map

Ψ S 0 : M k ⁢ ( Γ 0 ⁢ ( N ) , χ D ) → M k ⁢ ( ρ D )

restricted to the space

A k , S 0 ε 2 , … , ε r ⁢ ( Γ 0 ⁢ ( N ) , χ D )

is injective.

To demonstrate that the above construction is consistent with previous results, we consider the situation of Bruinier and Bundschuh [3] explicitly.

Example 6.20.

Let p > 2 be a prime and consider the case of D = A p t of even signature 1 - p ( mod 8 ) and level p. A set of inequivalent cusps is 𝔞 1 = ∞ = ( 1 : 0 ) and 𝔞 2 = 0 = ( 0 : 1 ) with widths w 1 = 1 and w 2 = p and cusp normalizers A 1 = I and A 2 = S . It is easy to verify, for instance using [17] or [13] that

ξ D ⁢ ( A 1 - 1 ) = 1 , ξ D ⁢ ( A 2 - 1 ) = e 8 ⁢ ( sign ⁡ ( D ) ) = e 8 ⁢ ( 1 - p ) = ε p , χ D = χ p = ( ⋅ p ) .

In this case, w 1 = 1 and w 2 = p , and so

w 2 ⁢ | D p | | D | = p .

It follows that if F ∈ M k ⁢ ( Γ 0 ⁢ ( p ) , χ p ) with Fourier expansions F i = ∑ n ≥ 0 b i ⁢ ( n ) ⁢ q n / w i , then

Φ { 0 } ∘ Ψ { 0 } ⁢ ( F ) = ∑ n ≥ 0 a ⁢ ( n ) ⁢ q n

has Fourier coefficients

a ⁢ ( n ) = b 1 ⁢ ( n ) + p ⁢ ε p ⁢ b 2 ⁢ ( p ⁢ n ) .

Recall the operators U p and W p defined by

U p ⁢ F = ∑ b ⁢ ( mod ⁢ p ) F | k ⁢ T b ,

W p F = F | S V p m ,

where

V p = ( p 0 0 1 / p ) .

It is easy to see that if F | W p U p = λ F , then b 2 ⁢ ( p ⁢ n ) = λ ⁢ p - 1 ⁢ b 1 ⁢ ( n ) , and therefore

a ⁢ ( n ) = b 1 ⁢ ( n ) ⁢ ( 1 + λ ⁢ ε p ⁢ p - 1 / 2 ) .

It can be shown that the eigenvalues of W p ⁢ U p are precisely λ = ϵ ⁢ ε ¯ p ⁢ p with ϵ = ± 1 and it is clear that a ⁢ ( n ) = b 1 ⁢ ( n ) ⁢ ( 1 + ϵ ) , so the lifting map is an isomorphism when restricted to the eigenspace corresponding to eigenvalue ϵ = 1 . This is of course exactly the same result as in [3] and the subspace A S 0 ε 2 ⁢ ( Γ 0 ⁢ ( p ) , χ D ) is the same as A k ϵ ⁢ ( p , χ p ) .

Explicit versions of Theorem 5.5 have been obtained for square-free levels by Scheithauer [14], for positive fundamental discriminants by Zhang [20] and for cyclic anisotropic discriminant forms with D ⁢ [ 2 ] = A 2 t also by Zhang [21]. The latter method can be extended to an arbitrary number of constituents of the form A 2 t in the case of S 0 = { 0 } , but in the general case we also need to consider the factors

| c i - 1 ⁢ S 0 / D c i | ⁢ | D c i * ∩ S 0 | .

7 Applications and examples

The original motivation to study these lifts in [13, 12] was to study automorphic products with certain properties. The explicit examples of lifts for integer weight Eisenstein series and eta products were used to provide general classes of examples of vector-valued modular forms for Weil representations that were then further lifted to automorphic products.

The main difficulty with applying the lifting maps to an arbitrary modular form F is that we need to know the Fourier expansion of F at every cusp. This is in general only the case a priori for Eisenstein series and eta quotients, the latter being used extensively by Scheithauer in [13, 14] to construct examples.

For integer weight modular forms, this can also be achieved if the level is square-free by using Atkin–Lehner involutions. For more general levels, it is possible to find these expansions using computational methods as, e.g., introduced in [7] using sums of products of Eisenstein series. Similar difficulties appear in the case of half-integral weight, and one way to overcome this is to use a restriction to certain symmetric subspaces as in [20, 21].

To illustrate the method, we will first give the general construction in the simplest case of Γ 0 ⁢ ( 4 ) , corresponding to a finite quadratic module A 2 t with t ≡ 1 or 3 ( mod 4 ) .

For more details of how to do this type of calculation in practice, see the repository [18] which contains general algorithms for finite quadratic modules and Weil representations as well as examples of lifting maps including the example given below.

7.1 Liftings from Γ 0 ⁢ ( 4 )

Let D = A 2 t with signature 1 and level N = 4 . Choose 𝔞 1 = ∞ = ( 1 : 0 ) , 𝔞 2 = ( 0 : 1 ) and 𝔞 3 = ( 1 : 2 ) as representatives of the cusps of Γ 0 ⁢ ( 4 ) and choose normalizers:

A 1 = I = ( 1 0 0 1 ) , A 2 = ( 0 - 1 1 0 ) , A 3 = ( 1 0 2 1 ) .

The widths of the cusps are w 1 = 1 , w 2 = 4 and w 3 = 1 , and the associated stabilizers are

T 1 = T = ( 1 1 0 1 ) , T 2 = ( 1 0 - 4 1 ) , T 3 = ( - 1 1 - 4 3 ) .

It is easy to see that χ D = v θ t in this case, and then χ D ⁢ ( T 1 ) = 1 , χ D ⁢ ( T 2 ) = 1 and χ D ⁢ ( T 3 ) = i t = e ⁢ ( t 4 ) . Hence, α ∞ = α 0 = 0 and α 1 2 = t 4 .

Since the group is isomorphic to ℤ / 2 ⁢ ℤ ≃ { 0 , 1 2 } ( mod ℤ ) , there is a unique isotropic subgroup S 0 = { 0 } and it is easy to see that D 0 ⁣ * = { 0 } , D 1 ⁣ * = D and D 2 ⁣ * = { 1 2 } . Furthermore, D 0 = D 2 = D and D 1 = { 0 } . Recall that Q c ⁢ ( x c + c ⁢ y ) = c ⁢ Q ⁢ ( y ) and x c = 0 unless 2 ∥ c , in which case x c = 1 2 . Hence,

Q 0 ⁢ ( 0 ) = 0 , Q 1 ⁢ ( 0 ) = 0 , Q 1 ⁢ ( 1 2 ) = Q - 1 ⁢ ( 1 2 ) = Q ⁢ ( 1 2 ) = 1 4 , Q 2 ⁢ ( 1 2 ) = Q - 2 ⁢ ( 1 2 ) = 0 .

Let F ∈ ℳ k ⁢ ( Γ 0 ⁢ ( 4 ) , v θ 2 ⁢ k ) and assume that F | A i ( τ ) = ∑ b i ( n ) q n is the Fourier expansion of F at the cusp 𝔞 i . It is easy to see that f = f S 0 , F ∈ ℳ k ⁢ ( ρ D ) , where D = A 2 1 or A 2 3 , depending on whether 2 ⁢ k ≡ 1 or 3 ( mod 4 ) . If we write f = ∑ α ∈ D f α ⁢ 𝐞 α , then the Fourier coefficients of f α are given by

c ⁢ ( α , n ) = ∑ i = 1 κ Λ i ⁢ ( α ) ⁢ b i ⁢ ( n ⁢ w i ) ,

where

Λ i ⁢ ( α ) = | D c i | | D | ⁢ ξ ⁢ ( A i - 1 ) ⁢ w i ⁢ ∑ ω ∈ D c i ⁣ * ∩ D S 0 c i ⁣ * ∩ { α } e ⁢ ( - d i ⁢ Q c i ⁢ ( ω ) ) .

Using the formulas from [17] and the correction from Remark 3.4, it follows that if D = A 2 t and A = ( a b c d ) ∈ SL 2 ⁡ ( ℤ ) with c ≠ 0 , then

ξ ⁢ ( A ) = ( - a c ) ⁢ ( - a , c ) ∞ ⁢ ( - 1 , c ) ∞ ⁢ e 8 ⁢ ( - 2 ⁢ t ) ⁢ { ( c ⁢ t 2 ) ⁢ e 8 ⁢ ( c ⁢ t ) , c ⁢ odd, ( a 2 ) ⁢ e 8 ⁢ ( ( c 2 + t - 1 ) ⁢ ( a + 1 ) ) , c ⁢ even.

It follows now that

ξ ⁢ ( A 1 ) = 1 ,

ξ ⁢ ( A 2 - 1 ) = ξ ⁢ ( 0 1 - 1 0 ) = ( 0 , - 1 ) ∞ ⁢ ( - 1 , - 1 ) ∞ ⁢ ( t 2 ) ⁢ e 8 ⁢ ( - 3 ⁢ t ) = ( t 2 ) ⁢ e 8 ⁢ ( t ) = ( t 2 ) ⁢ i t ,

ξ ⁢ ( A 3 - 1 ) = ξ ⁢ ( 1 0 - 2 1 ) = ( - 1 - 2 ) ⁢ ( - 1 , - 2 ) ∞ ⁢ ( - 1 , - 2 ) ∞ ⁢ e 8 ⁢ ( - 2 ⁢ t ) ⁢ ( 1 2 ) ⁢ e 8 ⁢ ( 2 ⁢ ( t - 2 ) ) = ( - 1 ) ⁢ e 8 ⁢ ( - 4 ) = 1 .

One can now see that Λ i ⁢ ( α ) = 0 unless α ∈ D c i ⁣ * , and otherwise

Λ i ⁢ ( α ) = | D c i | | D | ⁢ ξ ⁢ ( A i - 1 ) ⁢ w i ⁢ e ⁢ ( - d i ⁢ Q c i ⁢ ( α ) ) .

The non-zero elements are

Λ 1 ⁢ ( 0 ) = 2 2 ⁢ ξ ⁢ ( I ) ⁢ e ⁢ ( - Q - 1 ⁢ ( 0 ) ) = 1 ,

Λ 2 ⁢ ( 0 ) = 4 ⁢ 1 2 ⁢ ξ ⁢ ( A 2 - 1 ) = 2 ⁢ 2 ⁢ ( t 2 ) ⁢ i t ,

Λ 2 ⁢ ( 1 2 ) = 4 ⁢ 1 2 ⁢ ξ ⁢ ( A 2 - 1 ) ⁢ e ⁢ ( - 0 ⋅ Q - 1 ⁢ ( 1 2 ) ) = 2 ⁢ 2 ⁢ ( t 2 ) ⁢ i t ,

Λ 3 ⁢ ( 1 2 ) = 2 2 ⁢ ξ ⁢ ( A 3 - 1 ) ⁢ e ⁢ ( - 1 ⋅ Q - 2 ⁢ ( 1 2 ) ) = 1 ⋅ e ⁢ ( 0 ) = 1 .

Note that if the weight is k + 1 2 , we can choose t = 2 ⁢ k + 1 ( mod 8 ) , and then

( 2 t ) = ( 2 2 ⁢ k + 1 ) and i t = i k + 1 / 2 .

So if we set

ε k = ( 2 2 ⁢ k + 1 ) ⁢ i k + 1 / 2 ,

then the map

Ψ { 0 } : M k + 1 / 2 ⁢ ( Γ 0 ⁢ ( 4 ) ) → ℳ k + 1 / 2 ⁢ ( ρ A 2 t )

is given in terms of Fourier coefficients as

c ⁢ ( 0 , n ) = b ∞ ⁢ ( n ) + 2 ⁢ 2 ⁢ ε k ⁢ b 0 ⁢ ( 4 ⁢ n ) , n ∈ ℤ ,

c ⁢ ( 1 2 , n ) = 2 ⁢ 2 ⁢ ε k ⁢ b 0 ⁢ ( 4 ⁢ n ) + b 1 2 ⁢ ( n ) , n ∈ ℤ + 1 4 .

To illustrate the injectivity discussed in Section 6, we consider the explicit inverse lift from Proposition 5.5. Writing Φ 0 and Ψ 0 instead of Φ { 0 } and Ψ { 0 } , we have

Φ 0 : F ↦ F 0 ,

and the composition with Ψ 0 is then simply

Φ 0 ∘ Ψ 0 ⁢ ( F ) = ∑ n ∈ ℤ ( b ∞ ⁢ ( n ) + 2 ⁢ 2 ⁢ ε k ⁢ b 0 ⁢ ( 4 ⁢ n ) ) ⁢ q n .

Following the construction by Kohnen in [8] and setting L = U 4 ⁢ W 4 , it is easy to see that L ⁢ F = λ ⁢ F if and only if

b 0 ⁢ ( 4 ⁢ n ) = λ ⁢ ( 2 ⁢ i ) - k - 1 / 2 ⁢ b ∞ ⁢ ( n ) ,

in other words

Φ 0 ∘ Ψ 0 ⁢ ( F ) = ( 1 + 2 1 - k ⁢ ( 2 2 ⁢ k + 1 ) ⁢ λ ) ⁢ F .

It was shown by Niwa [10] that L is Hermitian and has eigenvalues

α 1 = 2 k ⁢ ( 2 2 ⁢ k + 1 ) and α 2 = - 2 k - 1 ⁢ ( 2 2 ⁢ k + 1 ) ,

and hence Φ 0 ∘ Ψ 0 ⁢ ( F ) = 3 ⁢ F if L ⁢ F = α 1 , and Φ 0 ∘ Ψ 0 ⁢ ( F ) = 0 if L ⁢ F = α 2 . In particular, this shows that Ψ 0 restricted to the Kohnen plus-space is an isomorphism (onto the image). Note that this case is also covered by Zhang [21].

For an explicit example, we need to know the coefficients at all cusps and as a simple example we choose the classical Jacobi theta function.

Example 7.21.

Consider the Jacobi theta function

θ ⁢ ( τ ) = ∑ n = - ∞ ∞ q n 2 = ∑ n ∈ ℤ b ∞ ⁢ ( n ) ⁢ q n = 1 + 2 ⁢ q + 2 ⁢ q 4 + 2 ⁢ q 9 + 2 ⁢ q 16 + 2 ⁢ q 25 + 2 ⁢ q 36 + ⋯ .

The Fourier expansions of θ at the cusps at 0 and 1 2 are given by

θ 0 ⁢ ( τ ) = θ | S ⁢ ( τ ) = ∑ n ∈ ℤ b 0 ⁢ ( n ) ⁢ q n 4 ,

θ 1 2 ⁢ ( τ ) = θ | A 3 ⁢ ( τ ) = ∑ n ∈ ℤ + 1 4 b 1 2 ⁢ ( n ) ⁢ q n .

The explicit values of the coefficients

b 0 ⁢ ( n ) = ( - i / 2 ) ⁢ b ∞ ⁢ ( n ) , n ∈ ℤ ,

b 1 2 ⁢ ( n ) = b ∞ ⁢ ( 4 ⁢ n ) , n ∈ ℤ + 1 4 ,

can be found using related theta functions or the eta quotient identity

θ ⁢ ( τ ) = η 5 ⁢ ( 2 ⁢ τ ) η 2 ⁢ ( τ ) ⁢ η 2 ⁢ ( 4 ⁢ τ ) .

The lifted function is therefore given by Ψ 0 ⁢ ( θ ) = ∑ γ f γ ⁢ 𝐞 γ , where

f γ = ∑ n ∈ ℤ + Q ⁢ ( γ ) c ⁢ ( γ , n ) ⁢ q n .

Further, if we define ε 0 = i 1 / 2 , then

c ⁢ ( 0 , n ) = b ∞ ⁢ ( n ) + 2 ⁢ 2 ⁢ ε 0 ⁢ b 0 ⁢ ( 4 ⁢ n ) = b ∞ ⁢ ( n ) + 2 ⁢ b ∞ ⁢ ( 4 ⁢ n ) = 3 ⁢ b ∞ ⁢ ( n ) , n ∈ ℤ ,

c ⁢ ( 1 2 , n ) = 2 ⁢ 2 ⁢ ε 0 ⁢ b 0 ⁢ ( 4 ⁢ n ) + b 1 2 ⁢ ( n ) = 2 ⁢ b ∞ ⁢ ( n ) + b ∞ ⁢ ( 4 ⁢ n ) = 3 ⁢ b ∞ ⁢ ( n ) , n ∈ ℤ + 1 4 ,

where we also used that b ∞ ⁢ ( 4 ⁢ n ) = b ∞ ⁢ ( n ) for all n.

8 Further work

The aim of the current paper has been to extend the fundamental techniques, and in particular the liftings from Γ 0 ⁢ ( N ) , used by Scheithauer in [13, 14] to arbitrary even integral lattices. Many of the ideas developed in those two papers should extend to the general case and it would be particularly interesting to study the construction of lifts from Γ 1 ⁢ ( N ) and Γ ⁢ ( N ) and their properties in more detail.

Communicated by Jan Bruinier

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/V026321/1

Funding statement: This work was partially supported by the Engineering and Physical Sciences Research Council (No. EP/V026321/1).

Acknowledgements

I would like to thank the anonymous referee for their patience and many helpful comments, and Nils Scheithauer for clarifying details of [13, 14] during my time in Darmstadt when this work was initiated.

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Received: 2022-11-28

Revised: 2023-07-04

Published Online: 2023-08-31

Published in Print: 2024-01-01

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https://doi.org/10.1515/forum-2022-0353

Keywords for this article

Modular forms; Weil representations; finite quadratic modules; liftings

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