Taylor expansions of Jacobi forms and linear relations among theta series

We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated with the $A_2$ lattice. We present the complete SageMath code for the $D_4$ lattice.

1. Introduction 1.1. Taylor expansions of Jacobi forms. The concept of Jacobi forms of lattice index, or equivalently, Jacobi forms on H × C n (H denotes the upper half plane), was first investigated by Gritsenko [Gri88], where the author studied their relationship to modular forms on the orthogonal group of a quadratic space of signature (2, n), and the action of Hecke operators on them. This generalized the theory of classical Jacobi forms developed by Eichler and Zagier [EZ85]. A further generalization to Siegel-Jacobi forms on H j × C j×n was obtained by Ziegler [Zie89] in the spirit of Eichler and Zagier. The present paper focuses on Jacobi forms of lattice index, so by the term "Jacobi forms" we mean forms of lattice index. Ajouz, under supervision of Skoruppa and Walling, developed systematically the Hecke theory of such forms, and the relation to elliptic modular forms, in his doctoral thesis [Ajo15].
Ajouz investigated liftings to and from elliptic modular forms via many tools, such as a generalization of Skoruppa-Zagier liftings [SZ88], Shimura correspondence ( [Shi73], [Niw75] and [Cip83]), stable isomorphisms between lattices ( [Nik79]), theta decompositions and Weil representations ([Ajo15, §2.3, §2.4]). On the contrary, in the present paper, we shall investigate the relation between Jacobi forms and elliptic modular forms via Taylor expansions with respect to the point 0 ∈ C n . In this direction, there are many excellent works. Eichler and Zagier has worked out the scalar index case. Ibukiyama [Ibu12] investigated Taylor expansions of Jacobi forms on H j × C j×1 . Bringmann, Mahlburg, and Rhoades [BMR14] considered those of mock-Jacobi forms. Later Bringmann [Bri18] considered those of non-holomorphic Jacobi forms. More recently, Ittersum [vI21], motivated by the problem of finding complex-valued functions on all partitions that are quasimodular forms for congruence subgroups, developed a theory of Taylor expansions of strictly meromorphic quasi-Jacobi forms.
We describe briefly our main result (Theorem 6.2). For any weight (integral or half-integral), any integral lattice (even or odd), any finite index subgroup of the modular group, and any representation with a finite index kernel, we obtain embeddings of the space of Jacobi forms (vector-valued or scalar-valued) into the direct product of finite number of spaces of elliptic modular forms. A key point is how to choose these spaces of modular forms, for which we give a computational criterion (Proposition 6.9, realized as a SageMath program [Sag21]).
To avoid overlapping the main text, we illustrate the main theorem with a typical example (scalar-valued and integral-weight). Let L = Z 2 , equipped with a quadratic form 1 Q(x, y) = 1 2 x 2 + xy + 2y 2 . The bilinear form associated with this quadratic form is B((x 1 , y 1 ), (x 2 , y 2 )) = Q(x 1 + x 2 , y 1 + y 2 ) − Q(x 1 , y 1 ) − Q(x 2 , y 2 ). The Gram matrix is G B = ( 1 1 1 4 ), so Q(x, y) = 1 2 (x, y)G B (x, y) T . Put L = (L, B). Let k be a positive integer, G be a finite index subgroup of SL 2 (Z), and χ : G → C × be a group character whose kernel is of finite index. The space J k,L (G, χ), by definition, consists of all holomorphic functions φ(τ, z) on H × C 2 subject to the following conditions: (1) For any ( a b c d ) ∈ G, (2) For any v, w ∈ L, φ (τ, z + τ v + w) = e (−τ Q(v) − B(v, z)) φ(τ, z), (3) For any ( a b c d ) ∈ SL 2 (Z), we have c(n, t)e (nτ + B(t, z)) , where c(n, t) ∈ C, and the series converges absolutely. The notation e (x) means exp(2πix), and L ♯ denotes the dual lattice of L. The reader may also compare this definition to [GSZ19,§9,Definition], and should notice a subtle difference: there is no factor e (Q(v + w)) in the second condition in our definition, but there is such one in [GSZ19]. (As a side effect, in [GSZ19], the variable t in c(n, t) ranges over L ♯ ev , not L ♯ .) Both definitions make sense, but they correspond to different spaces! For the definition in broad generalities, including the case of half-integral weights, of even or odd lattices, and of scalar-valued or vector-valued forms, see Definition 2.3 and 2.6. Put q n = e (nτ ) for tradition. Let M k (G, χ) and S k (G, χ) denote the space of modular forms, and that of cusp forms of weight k on the group G with character χ respectively. Then our main result, for this special case, says that: Theorem (A special case of Corollary 6.7). The map is a C-linear embedding, where φ(τ, z) = n,t c(n, t)q n e (B(t, z)) , D 0 (φ) = n t c(n, t) q n , D 1 (φ) = n t c(n, t) · (t 1 + 4t 2 ) q n , D 2 (φ) = n t c(n, t) · k(t 1 + 4t 2 ) 2 − 4n q n .
It is proper in this place to explain the relation of "Taylor expansions" in the title and the main theorem: each coefficient of terms of D 0 (φ), D 1 (φ), D 2 (φ) is some modified Taylor coefficient of φ.
We develop necessary tools to prove the main theorem from Section 2 to 5. In Section 2, we recall basic definitions and properties of Jacobi forms in broad generalities. In Section 3, we develop some operators used to define the sequence of operators D 0 (φ), D 1 (φ) and so on. To get rid of the problem of convergence, we work with formal Laurent series of several variables which transform like modular forms. At the end of this section, we obtain operators that map formal Laurent series of low weights to series of high weights. See Proposition 3.21 for details. Note that results in this section may be of independent interest. In Section 4, we describe operators that map Jacobi forms to modular forms. It turns out that for each vector p ∈ Z n ≥0 , where n is the number of elliptic variables, there is an operator D k,p that maps Jacobi forms of weight k to elliptic modular forms of weight k + s(p), where s(p) is the sum of p's components. See Theorem 4.6 for details. As would be discussed in Remark 4.7, this theorem belongs to van Ittersum. We reprove it in a different approach. Then we give the q-expansion of each operator D k,p acting on Jacobi forms (Proposition 4.11), which plays an important role in the following. At the end of this section (Proposition 4.15), we prove the direct product of all these operators, gives an isomorphism from all formal Laurent series which transform like modular forms, onto the direct product of infinitely many spaces of modular forms, hence gives an embedding of the space of Jacobi forms, into the same codomain. This proof is in the spirit of Eichler and Zagier (compare it to [EZ85,p. 34]). Now for proving the main theorem, it remains to show how to choose finitely many operators such that the direct product is still injective, for which we need the technique of theta decompositions. Thus, in Section 5 we review the theory of Jacobi theta series of lattice index, Weil representations and theta decompositions. We prove a version of theta decompositions (Theorem 5.2) which serves our purpose. The rest of this section is devoted to discussing the relation between theta decompositions and Taylor expansions. After these preparations we state and prove the main theorem in Section 6. The finitely many operators D k,p are choosen in such way that in the theta decomposition of any Jacobi form φ, D k,p φ = 0 for all these p could imply the theta coefficients of φ all vanish, and hence φ itself vanishes. The rest of this section contains some immediate corollaries dealing with lattices of determinant 2 and 3, and a computational criterion (Proposition 6.9) for deciding what p's to choose.
As an application of the main theorem (Theorem 6.2), we give an algorithm for searching and proving this type of identities, not only for the A 2 lattice, but for arbitrary lattice in principle. More precisely, for any even integral positive definite symmetric bilinear form B on L = Z n (extended to C n by bilinearity), and suitably choosen α and β, we could find all linear relations among functions θ N ′ α,β , where N ′ is a fixed positive integer dividing the level of (L, B) (the least positive integer N such that N · Q(x) ∈ Z for all x ∈ Z n ). The algorithm roughly says the following.
For the precise version (and definitions of concepts not introduced yet), see Theorem 7.10. For the definition of the polynomial P p , see Definition 4.10.
D 4 2 4 4 258 Theorem 7.11 A 2 3 3 3 99 Theorem 7.12 A 3 2 2 4 635 Theorem 7.13 7.9, we answer the question for which α and β, functions θ N ′ α,β are in the same space, that is, with the same character. Immediately after that, we state and prove the main theorem of this section, Theorem 7.10. The remaining of this section is divided into five subsections. Each of the first three subsections deals with one lattice in Table 1. In §7.4, we present a non-trivial example showing that, in Theorem 6.2, we can not choose the set of p's with small cardinality. In §7.5, we show that the third (fifth) powers of nine (twenty-five) functions θ α,β associated with the quadratic form x 2 + xy + 4y 2 are linearly independent.
In Appendix A, we present the complete SageMath source code used to produce and check linear relations among series for the D 4 lattice, along with detailed discussion on the usage and on how to fit the program for other lattices.
1.3. Notations. The symbols Z, Q, R, C denote the ring of integers, rationals, reals and complex numbers respectively. The symbol R >n (R ≥n resp.) refers to the set of elements in R that are greater than (or equal to) n, The symbol C × denotes the multiplicative group of nonzero complex numbers. The symbol P 1 (Q) denotes the set of projective lines over Q (Q-subspaces of dimension one), whose elements can be represented by formal fractions a/b.
For a commutative ring R (with multiplicative identity), the general linear group GL 2 (R) is the group of 2×2 matrices over R with invertible determinant, and the special linear group SL 2 (R) is the subgroup of GL 2 (R) consisting of matrices of determinant 1. If R is a subring of R, then the symbol GL + 2 (R) denote the subgroup consisting of matrices of positive determinant. For a subgroup G of GL + 2 (R), G denote its double cover (see the second paragraph of Section 2), and G denotes G/{±I} if −I ∈ G, or G itself if −I / ∈ G, where I = ( 1 0 0 1 ). We put PSL 2 (Z) = SL 2 (Z).
Throughout this paper, V always denotes a linear space of dimension n ∈ Z ≥1 , and V its complexification C ⊗ R V . The upper half plane H, is the subset of C consisting of numbers with positive imaginary part. Besides V, the symbol W always denotes a complex linear space of dimension m ∈ Z ≥1 . These sets serve as domains and codomains of Jacobi forms considered here -a Jacobi form is implicitly assumed as a function from H × V to W. See the first paragraph of Section 2. By C(H × V, W), O(H × V, W), we mean the set of continuous maps, and that of holomorphic maps from H × V to W. By GL(W), we mean the group of non-singular linear operators on W.
The symbols H(V ), H(L, n) denote some kind of Heisenberg groups. Refer to the paragraph following the proof of Lemma 2.2 for details. The symbol L usually denotes a Z-lattice in V , which is assumed to be integral at most time, and L means L equipped with a bilinear form B (i.e., L = (L, B)). By L ♯ , or more precisely, L ♯ , we mean the dual lattice of L (see the paragraph followed by Proposition 2.4). For a positive integer N, the notation L N denotes the lattice with the same underlying Z-module L, but a different bilinear form N · B.
For simplicity, put e(z) = exp 2πiz with z ∈ C, and q n = e (nτ ) with τ ∈ H, n ∈ Q. The notation Γ(z) refers to the Euler Γ-function. For two maps f 1 and f 2 , f 1 • f 2 means the map that sends x to f 1 (f 2 (x)), provided that the codomain of f 2 coincides with the domain of f 1 . For x ∈ R, the floor [x] equals the largest integer not exceeding x. For two groups G and H where H is a subgroup of G, [G : H] means the index of H in G. For a nonzero integer d, d −1 = sgn (d) denotes the sign of d, and 0 −1 = 1 by convention. Finally, for notations concerning formal Laurent series, refer to first two paragraphs of Section 3.

Jacobi forms of lattice index
We recall basic concepts and properties concerning Jacobi forms of lattice index. Throughout this paper, the symbol V always denotes a finite dimensional real vector space (whose dimension will be denoted by n), and V its complexification, i.e., V = V ⊕ iV . Instead of v + iw, we use the notation [v, w] to represent elements in V, where v, w ∈ V .The symbol W denotes another finite dimensional complex vector space of complex dimension m. The functions studied here are holomorphic ones from H × V to W, where H denotes the upper half plane of the complex numbers. Let B : V × V → R be a symmetric R-bilinear form and by abuse of language, B also denotes its C-bilinear extension to V × V. For any additive subgroup X of V, we write X to mean the group X equipped with the restriction of the form B to X × X, so V = (V, B). There is another "extension" of B on V × V to V × V, which is denoted by A and is defined by To state the modular and abelian transformation laws, and to define operators such as Hecke operators and double coset operators, we need a group, whose elements represent transformations of functions studied in this paper. The group used here is the semidirect product of a general linear group and a Heisenberg group, which we now explain in detail. Let GL + 2 (R) be the group of 2×2 real matrices with positive determinants, SL 2 (R) be the subgroup of matrices of determinant 1, and SL 2 (Z) be the subgroup of matrices with integeral entries in SL 2 (R). To deal with half-integer-weight modular forms, we shall use the double cover of GL + 2 (R), which is denoted by GL + 2 (R), whose elements Note that by the function z → z r , we mean z → exp r log z, where we choose the branch of log such that −π < ℑ log z ≤ π, and that ε (c ′ τ + d ′ ) 1 2 means a function on τ ∈ H. For an element γ = ( a b c d ) ∈ GL + 2 (R), we define j(γ, τ ) = c ′ τ + d ′ , and γ(τ ) = aτ +b cτ +d = a ′ τ +b ′ c ′ τ +d ′ . The composition of GL + 2 (R) is given by For any subgroup G of GL + 2 (R), the notation G means the preimage of G under the natural projection from GL + 2 (R) onto GL + 2 (R), so we have SL 2 (R) and SL 2 (Z). For matrix γ = ( a b c d ) ∈ GL + 2 (R), the symbol γ means ( a b c d ) , (cτ + d) composition law is given by The group GL + 2 (R) acts from right on H(V ) as follows: Lemma 2.1. The function J B satisfies the cocycle condition. That is to say, for γ 1 , γ 2 ∈ GL + 2 (R) ⋉ H(V ) and Z ∈ H × V, one has Proof. A tedious but straightforward calculation only using definitions.
In the above formula, k is a number called the weight, and | k,B is called the slash operator of weight k and form B.
Lemma 2.2. Suppose k ∈ 1 2 Z. Then the slash operater is a right group action; that is, for f ∈ C(H × V, W) and γ 1 , γ 2 ∈ GL + 2 (R) ⋉ H(V ), we have (f | k,B γ 1 ) | k,B γ 2 = f | k,B (γ 1 γ 2 ), and the identity element of GL + 2 (R) ⋉ H(V ) gives the identity map on C(H × V, W). Note that when we restrict the set of functions to C ∞ (H × V, W), or to O(H × V, W), this action still remains well-defined.
Proof. This follows from that j(γ, τ ) and J B (γ, Z) satisfy cocycle condition (The case of j(γ, τ ) is classical, and the case of J B (γ, Z) is in Lemma 2.1.), and that (2.1) is a left group action.
By a Z-lattice L in V , we mean a free Zmodule in V , with a Z-basis which is also a R-basis of V . As mentioned above, set L = (L, B| L×L ). We say that L is integral if B(L, L) ⊆ Z, and L is even if Q(L) ⊆ Z. If L is integral but not even, then we say L is odd.
Then H(L) is a subgroup of H(V ), and for any subgroup G of SL 2 (Z), the set G ⋉ H(L) is a subgroup of GL + 2 (R) ⋉ H(V ). If B(L, L) ⊆ Q, we can make the Heisenberg group a cover of L×L of finite index. To do this, define H(L, n) = {([v, w], ξ) ∈ H(L) : ξ 2n = 1} for n ∈ Z ≥1 . This is a subgroup when B(L, L) ⊆ 1 n Z. Particularly, if L is integral, then H(L, 1) is a subgroup, which is used to state the elliptic transformation laws for Jacobi forms. Though there is a subgroup of H(L) that is isomorphic to L × L when L is even, we shall not use this subgroup for the sake of dealing with even and odd lattices simultaneously. Note that H(L, 1) is abelian.
Convention 1. From now on, we always assume that L is an integral lattice, and G is a subgroup of SL 2 (Z). So G ⋉ H(L, 1) is a subgroup of GL + 2 (R) ⋉ H(V ). Moreover, k always denotes a half integer (or an integer), and the weight of Jacobi forms considered is assumed to be k, unless explicitly specified.
Definition 2.3. Let ρ : G ⋉ H(L, 1) → GL(W) be a group representation, and φ ∈ C(H×V, W). We say that φ transforms like a Jacobi form of weight k and index L under the group G with the representation ρ, if for any γ ∈ G ⋉ H(L, 1), we have φ| k,B γ = ρ(γ) • φ. Denote the space of such functions by J cont. k,L (G, ρ), and the subspace of holomorphic ones by J O k,L (G, ρ). Note that the slash operator and applying a linear operator on the left commute, i.e., (L • φ)| k,B γ = L • (φ| k,B γ) for any linear operator L on W. As a consequence, to show that a map φ transforms like a Jacobi form, it suffices to verify the tansformation laws corresponding to a set of generators of G ⋉ H(L, 1).
To make spaces of such (holomorphic) functions finite-dimensional, hence arithmetically interesting, we shall pose growth conditions at cusps of the quotient H × V modulo G ⋉ H(L, 1), which will be given by Fourier series. So we recall some fundamental facts about Fourier series on the n-torus, stated in a form suits us. Assume that B (on which turns out to be a lattice of the same rank if B is nondegenerate. To indicate the bilinear form, we also wrtie L ♯ .) On the other hand, there is an inner product on V induced by L, given by (f, g) = vol (F) −1 F f (x)g(x) dx, where f and g are complex-valued continuous L-periodic functions, and F is any closed fundamental domain being the preimage of [0, 1] n under a linear isomorphism from V onto R n which is simultaneously an isomorphism of L and Z n (recall that n = dim V ). This inner product (·, ·), is independent of the choice of F and the Lebesgue measure posed on V . A fundamental theorem in Fourier analysis asserts that any complexvalued continuous L-periodic function f on V can be decomposed as f (x) = t∈L ♯ a t e (B(t, x)) with a t ∈ C, where the equality means that the right-hand side converges unconditionally with repect to the inner product to the left-hand side. Put another way, the functions {x → e (B(t, x)) : t ∈ L ♯ } forms a maximal orthonormal basis of the space of complex-valued continuous L-periodic functions equipped with the inner product (·, ·). Since we deal mainly with holomorphic functions, the following proposition should be useful.
Then there exists a unique sequence c t for t ∈ L ′♯ , such that The series converges normally on V ′ ⊕ iD. The coefficients are given by where y ∈ D, the integral is with respect to the Lebesgue measure transformed from R d via the coordinate map from V ′ onto R d induced by the basis (e j ) 1≤j≤d , and F ′ is the inverse image of [0, 1] d under this coordinate map.
Proof. For the case of B ′ being positive definite, see [Fre11, Appendix to §6.8]. The general case follows from this special case.
By saying that a function series t f t defined on V ′ ⊕ iD converges normally, we mean that for any compact subset K of V ′ ⊕ iD, we have t sup x∈K |f t (x)| < ∞. This implies absolute convergence and compactly uniform convergence. But normal convergence is stronger. Note that this proposition generalizes immediately to functions taking values in finite-dimensional complex vector spaces. In this situation, in the definition of normal convergence, we shall substitute the absolute value by the norm. Since any norms on a finite-dimensional real or complex vector space are equivalent, we need not refer to the norm when talking about normal convergence. For us, the proposition applies to the case k,L (G, ρ) and γ ∈ SL 2 (Z)⋉H(L, 1), there exist a W-valued sequence c(n, t) with n ∈ 1 N 1 N 2 Z and t ∈ 1 c(n, t)q n e (B(t, z)) .
This series converges normally. As a consequence, it converges absolutely and locally uniformly at any (τ, z) ∈ H × V.
In practice, one need only check the expansion (2.2) for finitely many γ ∈ SL 2 (Z) ⋉ H(L, 1): Proposition 2.7. Use the notations and assumptions in Definition 2.6. Let {s i : 1 ≤ i ≤ w} be a system of representatives of the quotient . Then φ ∈ J k,L (G, ρ) if and only if φ| k,B γ i has an expansion of the form (2.2) with n − Q(t) ≥ 0 for each i. There are similar conclusions for J ! k,L (G, ρ), J weak k,L (G, ρ) and J cusp k,L (G, ρ). Still use the notations and assumptions in Definition 2.6. We give some properties on Fourier coefficients, some of which are used in the proof of Proposition 2.7. We use c γ (n, t), or more precisely c γ φ (n, t) to denote the coefficients c(n, t) in (2.2), to indicate that it depends on γ and φ. It is useful to extend the domain of the map (n, t) → c γ (n, t) to R × V , such that c γ (n, t) = 0 unless n ∈ 1 N Z and t ∈ 1 N L ♯ (N = N 1 N 2 ). Let v ∈ L and γ ∈ SL 2 (Z); applying the operator | k,B ([v, 0], 1) to φ| k,B γ on the right, computing the Fourier coefficients in two ways and using the uniquessness of Fourier coefficients yield that for any (n, t) ∈ R×V . A similar formula holds for γ ∈ SL 2 (Z)⋉H(L, 1).
Proof. The "only if" part is obvious, and the proofs for J ! k,L (G, ρ), J weak k,L (G, ρ) and J cusp k,L (G, ρ) are similar to that for J k,L (G, ρ). Hence we only prove the "if" part for J k,L (G, ρ). Suppose that φ| k,B γ i has an expansion of the form (2.2) with n − Q(t) ≥ 0 for each i, where the Fourier coefficient c(n, t) is denoted by c i (n, t) here, to indicate its dependence on γ i . Let , ξ) ∈ SL 2 (Z) ⋉ H(L, 1) be arbitrary, and put γ = ( a b c d ). We shall prove that φ| k,B (( a b c d ) , ε, [v, w], ξ) has an expansion of the form (2.2) with the property c(n, t) = 0 =⇒ n − Q(t) ≥ 0. Suppose γ(∞) is equivalent to s i modulo G, so there is some γ ′ ∈ G such that γ(∞) = γ ′ (s i ). Thus, γ −1 i γ ′−1 γ(∞) = ∞, and in consequence, there is some e ∈ Z and δ = ±1 such that γ = γ ′ γ i (δT e ), where T = ( 1 1 0 1 ). Hence, there is some Using the change of variables corresponding to the following bijection and applying (2.3), we obtain Now if n − Q(t) = n − Q(δt) < 0, then c i (n, δt) = 0, and hence c(n, t) = 0, which concludes the proof.
We conclude this section by recalling another basic property which would be used later, concerned with multiplication of two scalar-valued Jacobi forms of lattice index.
Suppose B 1 and B 2 are two positive definite symmetric R-bilinear forms on V . (As usual, Q i (x) = 1 2 B i (x, x) for i = 1, 2.) Then their sum B 1 + B 2 , defined pointwisely, is also positive definite symmetric. Let L ♯ i denote the dual lattice of L in the inner product space (V, B i ) (i = 1, 2), and L ♯ 3 the dual of L in (V, B 1 +B 2 ), where L is any Z-lattice in V , not necessary integral.
for any x ∈ V . Moreover, this t belongs to 1 N L ♯ 3 , and can be obtained by the following way. Let B be a R-basis of V , and (t 1 1 , . . . , t n 1 ), (t 1 2 , . . . , t n 2 ) be coordinate row vectors of t 1 , t 2 with respect to B, respectively. Let G 1 , G 2 be the Gram matrices of B 1 , B 2 with respect to the basis B, respectively. Then the coordinate row vector (t 1 , . . . , t n ) of t is uniquely determined by the following matrix equation Lemma 2.9. Use notations and assumptions in the last lemma. Put G ′ = G 1 (G 1 +G 2 ) −1 G 2 , then G ′ is a positive definite symmetric matrix. Moreover, let B ′ denote the bilinear form on V ×V whose Gram matrix with respect to B is G ′ , and put Q ′ (x) = 1 2 B ′ (x, x). Then we have As a consequence, (Q 1 + Q 2 )(t) ≤ Q 1 (t 1 ) + Q 2 (t 2 ).

Proof. A direct verification.
Proposition 2.10. Let k 1 , k 2 ∈ 1 2 Z and B 1 , B 2 be two positive definite symmetric R-bilinear forms on V . Let L be a Z-lattice in V that is integral both in (V, B 1 ) and (V, B 2 ). Let G be a finite index subgroup of SL 2 (Z) and χ i be a linear character on G ⋉ H((L, B i ), 1) whose kernel is of finite index, for i = 1, 2. It is required that χ i ([0, 0], ξ) = ξ. Let f 1 , f 2 : H × V → C be two functions. Then we have following conclusions.
(1) The map , 1) is a linear character, which is denoted by χ 1 * χ 2 . The notation χ 1 · χ 2 denotes the pointwise multiplication. ( Moreover, if one of f 1 and f 2 is a Jacobi cusp form, then so is Proof. The conclusion on χ 1 * χ 2 can be proved by the definition of linear characters. Just pay attention to the fact that underlying group compositions of H((L, B 1 ), 1), H((L, B 2 ), 1) and H((L, , from which the transformation laws (see Definition 2.3) follow. The requirement on Fourier expansions (see Definition 2.6) follows from Lemma 2.8 and Lemma 2.9.
Remark 2.11. There are also parallel conclusions for weakly holomorphic Jacobi forms, and for weak Jacobi forms.

Formal Laurent series which transform like modular forms
For the sake of freely defining differential operators used to construct embeddings of spaces of Jacobi forms into products of spaces of modular forms, we shift the focus from functions in O(H × V, W) to formal series whose coefficients are modular forms in O(H, W). This section is devoted to developing such a theory. Let T 1 , T 2 , . . . , T n be formal variables. Let j = (j 1 , j 2 , . . . , j n ) be in Z n and set T j = T j 1 1 T j 2 2 · · · T jn n . Use the symbol j∈Z n O(H, W)T j to denote the set of all formal series whose terms are of the form h j (τ )T j with h j ∈ O(H, W) for each j ∈ Z n . This set becomes a C-space, also a O(H, C)-module under the ordinary addition and scalar multiplication. The symbol j≫−∞ O(H, W)T j denotes the subspace of those series j∈Z n h j T j ∈ j∈Z n O(H, W)T j with the property there exists a j 0 ∈ Z n such that h j = 0 implies 2 j ≥ j 0 . Equipped with the ordinary multiplication, the subspace j≫−∞ O(H, W)T j is a C-algebra (associative and unitary). We also need the subspace j≥j 0 O(H, W)T j with j 0 being a fixed vector.
Next we shall define slash operators on formal series. For this purpose, it is necessary to introduce some nonstandard notations. The symbol R n×n denotes the set of all n × n matrices with entries in R. For G, Λ in R n×n , s(G) denotes the sum of all entries of G, and G Λ denotes the product of all g λ where (g, λ) ranges over corresponding entries in G and Λ. By Λ * ,j and Λ i, * , we mean the j-th column and i-th row of Λ respectively. Put π i (Λ) = s(Λ i, * ) + s(Λ * ,i ), and π(Λ) = (π 1 (Λ), . . . , π n (Λ)). We define with λ ranging over all entries of Λ.For j ∈ Z n , s(j) denotes the sum of all components of j, as in the matrix case. Under such notations, one has, for instance, Definition 3.1. Let B = (e 1 , e 2 , . . . , e n ) be an ordered R-basis of V, and let G B be the Gram matrix of B with respect to B (i.e. the (i, j)-th entry is B(e i , e j ).). Let j∈Z n h j T j be a formal series in We warn the readers that 0 0 = 1 as usual in the theory of power series. Note that the sum used to define g p is actually a finite sum, since for Λ with s(Λ) sufficiently large, we have h p−π(Λ) = 0. One can also verify that, the formal series p∈Z n g p T p is again in j≫−∞ O(H, W)T j , and if, in addition, j∈Z n h j T j ∈ j≥j 0 O(H, W)T j , then p∈Z n g p T p ∈ j≥j 0 O(H, W)T j too. We shall prove that this is a group action, for which the following technical lemma is useful.
n×n , and z 1 , z 2 ∈ C. We have Proof. By the binomial theorem, it suffices to prove that, for any nonnegative integer u, v satisfying u + v = s(λ), the following formula holds: (3.6) Let X i,j be n 2 indeterminants (1 ≤ i, j ≤ n) and let X be the matrix whose (i, j)-th entry is X i,j . It is trivial that Now evaluating the coefficients of X Λ in both sides of the above formula leads to (3.6).
Proposition 3.3. The slash operator given in Definition 3.1 is a right group action of Proof. The fact that identity element in SL 2 (R) gives the identity map on j≫−∞ O(H, W)T j is obvious. So we should prove that, for γ 1 , γ 2 ∈ SL 2 (R), and j∈Z n h j T j ∈ j≫−∞ O(H, W)T j , the following formula holds: Then for p ∈ Z n , the coefficient of T p in the left-hand side of (3.8) is For a fixed Λ ∈ Z ≥0 n×n , the subsum satisfying Λ 1 + Λ 2 = Λ in (3.9) is Using Lemma 3.2, we have Inserting this into (3.10), we find that (3.9) equals the coefficient of T p in the right-hand side of (3.8), form which (3.8) follows.
Definition 3.1 has an equivalent form, namely, where a m,n is the (m, n)-th entry of G B . Maybe this form is more natural, since it coincides with the slash operators on Jacobi forms. See the paragraph that precedes Lemma 2.2. In fact, it is from (3.11) and the power series expansion of the exponential function that we work out (3.5). We can also give an inverse formula expressing h j by g p .
Proposition 3.4. Let j∈Z n h j T j and p∈Z n g p T p be two formal series in j≫−∞ O(H, W)T j . Then (3.5) holds for any p if and only if the following relation holds for any j: Proof. (3.5) =⇒ (3.12). Note that (3.5) and (3.11) are equivalent. Hence we have Now expand the first factor in the right-hand side of the above identity as a power series of T 1 , . . . , T n and compare the coefficients of T j in both sides; the desired relation then follows.
(3.12) =⇒ (3.5). Put Then by the first part of this proposition (which we have proved), h j and g ′ p also satisfy relation (3.12) with g replaced by g ′ . By induction on s(p) one finds that g p = g ′ p , for any p ∈ Z n , from which (3.5) follows.
We need a formula dealing with taking successive derivatives of (3.12), which is important in the proof of Proposition 4.15, a main result of the next section.
then for any u ∈ Z ≥0 , we have Proof. Induction on u.
We now describe connections between slash operators on Jacobi forms introduced in Section 2 and that on formal series introduced in Definition 3.1. As above, let B = (e 1 , . . . , e n ) be an ordered R-basis of V . We define a C-linear embedding, denoted by emb we expand the function f (τ, z 1 e 1 + . . . z n e n ) as a power series of (z 1 , . . . , z n ) ∈ C n around zero. Then emb B (f ) is defined as the resulting power series with z i 's replaced by T i 's.
Proposition 3.6. For any γ ∈ SL 2 (R), the following diagram (of the category of complex vector spaces) commutes: Proof. This is a direct consequence of (3.11).
There are another two types of linear operators acting on formal series, besides slash operators, which are important in investigating the Taylor expansion of Jacobi forms. The first is the following: Proposition 3.8. Use notations in Definition 3.1 and put γ = (( a b c d ) , ε). We have The second one is the following. Note that E i,j denotes the matrix whose (i, j)-entry is 1 but all other entries are 0.
Definition 3.9. Let B and G B be as in Definition 3.1 and let a i,j be the This is a linear operator on j∈Z n O(H, W)T j . Sometimes we write L k instead of L B k,B when B and B are implicitly known.
Proposition 3.10. Let j∈Z n h j T j ∈ j≫−∞ O(H, W)T j , and γ ∈ SL 2 (R). Let B be as in Definition 3.1. Then we have . Then the coefficient of T p -term in the lefthand side of (3.14) is On the other hand, the coefficient of T p -term in the right-hand side of (3.14) is Hence (3.14) is equivalent to This formula can be proved, using the fact Here λ i,j means the (i, j)-entry of Λ. This concludes the proof.
Remark 3.11. The composition T −p •L B k,B is of particular interest when s(p) = 2. For one-dimensional case, that is, the case n = 1 (in this case p is the integer 2), this operator maps n∈Z h n T n . This is the modified heat operator introduced in the proof of [EZ85, Theorem 3.2], but differs by a factor −1.
After defining necessary operators and exploring their basic properties, we now introduce the main object in this section.
Definition 3.12. Let H be a subgroup of SL 2 (Z), and ρ : H → GL(W) a representation. Fix an ordered R-basis B of V . Let j∈Z n h j T j be a formal series in j≫−∞ O(H, W)T j . We say that j∈Z n h j T j transforms like a modular form under the group H of weight k with repre- . Remark 3.13. The relationship between Jacobi forms of lattice index and formal sereis which transform like modular forms are as follows. Recall notations in Convention 1 and 2, and recall emb B used in Proposition 3.6. Then the image of , which is a direct consequence of Proposition 3.6. Some subspaces of M B k,B (H, ρ) are also necessary for us. We define (3.18) The following are another subspaces, in which s 0 is any integer: (H, ρ). We call (3.17) and (3.20) plus spaces, (3.18) and (3.21) minus spaces. Now we investigate how the operators T −p and L B k,B act on above spaces.
Proposition 3.14. Use notations in Definition 3.12. Let p and j 0 be two vectors in Z n , and let s 0 be an integer. Then , then T −p maps plus (minus resp.) spaces bijectively onto corresponding plus (minus resp.) spaces. If 2 ∤ s(p), then T −p maps plus (minus resp.) spaces bijectively onto corresponding minus (plus resp.) spaces.
Proof. It's a consequence of Proposition 3.8.
The above two propositions deal with general mapping properties of T −p and L B k,B acting on formal series which transform like modular forms, while the following is a special but crucial one for our purpose. Proof. We have known from the last proposition that (H, ρ). Then it follows from the expression of g p in Definition 3.9 that, s(p) = 0 implies g p = 0. Therefore, p∈Z n g p T p ∈ M B, 2,+ k,B (H, ρ), which concludes the proof. This is why we are interested in operators of the form T −p • L B k,B , with s(p) = 2, as announced in Remark 3.11. By composing several such operators, we obtain certain weight-raising operators.
Corollary 3.17. Let λ ∈ Z ≥1 . Let p 0 , . . . , p λ−1 be λ vectors in Z n with s(p i ) = 2, and let p −1 be a vector in Z n such that s(p −1 ) = 1. Then the operator , and the operator Proof. The first assertion follows from Proposition 3.14 and Lemma 3.16. The second assertion follows from the first one and the fact that We want to find explicit descriptions of operators (3.22) and (3.23). Luckily, there exists a simple expression.
Proposition 3.18. Fix the bilinear form B and the weight k as in Convention 1 and 2, and fix an ordered R-basis B of V as before. Suppose j∈Z n h j T j ∈ j∈Z n O(H, W)T j . Then the coefficient of T jterm of (3.22) acting on j∈Z n h j T j is where p = 0≤i<λ p i . On the other hand, the coefficient of T j -term of (3.23) acting on j∈Z n h j T j is the same as the last expression, but with p = −1≤i<λ p i and the Γ-factor replaced by Proof. The second assertion follows immediately from the first one, of which we give a sketch of proof. We use induction on λ. The case λ = 1 is clear using definitions. Assume that the case λ has been proved, and proceed to prove the case λ + 1. By induction hypothesis, what we should prove is where a i,j is the (i, j)-entry of G B . This could be proved by using a change of variable Λ ′ = Λ + E i,j in the second sum in the left-hand side, and then a straightforward calculation.
It follows that the operator (3.22) and (3.23) depend only on the sum of all p i 's, not on any individual p i . This leads to the following definition.
Definition 3.19. Let B, k, and B be as in Proposition 3.18. Suppose p is a vector in Z n with s(p) ≥ 0. We define an operator D B k,B,p , or simply D k,p , on j∈Z n O(H, W)T j as follows: (1) If s(p) = 0, 1, then D k,p = T −p .
(2) If s(p) ≥ 2 and 2 | s(p), then D k,p is defined to be (3.22) with p 0 , . . . , p λ−1 being vectors in Z n such that s(p i ) = 2 and Remark 3.20. By Proposition 3.18, the operator D k,p is well-defined. In fact, that proposition also gives explicit expression of each term of D k,p ( j∈Z n h j T j ) when s(p) ≥ 2. One can see that, this explicit expression for s(p) ≥ 2 is also valid for s(p) = 0, 1 (in this case, λ = 0). Finally note that, in the explicit expression, λ always equals We restate a useful proposition: Proposition 3.21. Let H be a subgroup of SL 2 (Z), and ρ : H → GL(W) a representation. Let k, B and B be as in Proposition 3.18.
The case s(p) = 0 or 1 is a special case of Proposition 3.14, while the other case is just a restatement of Corollary 3.17.
We conclude this section by explaining why these operators are relevant to Taylor expansions of Jacobi forms of lattice index. The Taylor coefficients of a Jacobi form φ ∈ J O k,L (G, ρ) with respect to some basis B are just the coefficients h j 's of emb B (φ) (See Remark 3.13). Hence each coefficient of D k,p (emb B (φ)) is a finite linear combination of Taylor coefficients of φ. In this way, we can regard each coefficient of D k,p (emb B (φ)) as certain kind of modified Taylor coefficient. Some of these modified Taylor coefficients are ordinary modular forms, which will be investigated in the next section.

Connections with modular forms
Recall some notations for modular forms. Let H be a subgroup of SL 2 (Z) and k be a half integer or integer. Let ρ : H → GL(W) be a group representation. Then the notation M O k (H, ρ) denotes all holomorphic functions h ∈ O(H, W) satisfying modular transformation equations on H with representation ρ and of weight k. If H is of finite index in SL 2 (Z), then the subspace of those functions meromorphic at all cusps of H is denoted by M ! k (H, ρ), and that holomorphic at all cusps is denoted by M k (H, ρ). Moreover, S k (H, ρ) denotes the space of h ∈ M k (H, ρ) which vanishes at all cusps of H. We omit the information W in these notations as in Section 2, since it can be recovered from ρ.
Proof. This follow immediately from Definition 3.1 and Definition 3.12.
If j∈Z n h j T j ∈ j∈Z n O(H, W)T j , we call h 0 the constant term. Combining the operation "taking the constant term" and D k,p introduced in the last section, we can construct higher weight modular forms.
Definition 4.2. Let k, B, B, H and ρ be as in Definition 3.12. Suppose p ∈ Z n with s(p) ≥ 0. We define an operator On the other hand, if 2 ∤ s(p), then D k,p ( j∈Z n h j T j ) equals In each case, λ = [s(p)/2].
Proof. A special case of Proposition 3.18 with j = 0. (k + s).
We now apply what have been achieved to Jacobi forms of lattice index, answering the question how to combine Taylolr coefficients of such forms to construct ordinary modular forms. Recall notations in Convention 1 and 2.
Remark 4.7. We must mention that, this theorem belongs to van Ittersum. See [vI21, Corollary 2.44]. He developped such a theorem for quasi-Jacobi forms in several elliptic variables. Our D k,p ( j∈Z n h j T j ) is the same as van Ittersum's ξ p (φ), up to a factor depending on k and p. The purpose that we state and reprove this theorem here is twofold. One reason is that our method of the proof is different from van Ittersum's, and the other is that this theorem is the beginning of our theory on Taylor expansions of Jacobi forms of lattice index.
Proof. By abuse of language, we write D k,p (φ), instead of the more heavy but precise notation D k,p (emb B (φ)). Then we should prove D k,p (φ) ∈ M ! k+s(p) (G, ρ| G ). We have already known that D k,p (φ) ∈ M O k+s(p) (G, ρ| G ) by definition. Hence, it remains to show that, for any γ ∈ SL 2 (Z), the q-expansion of D k,p (φ)| k+s(p) γ has only finitely many terms of negative power of q, where | k+s(p) γ is the usual slash operator on modular forms. Note that D k,p (φ)| k+s(p) γ = D k,p (φ| k,B γ) by Proposition 3.6, 3.8 and 3.10. By the definition of weakly holomorphic Jacobi forms (Definition 2.6), there exists an n 0 ∈ Z such that (2.2) holds with c(n, t) = 0 =⇒ n ≥ n 0 . By absolute convergence, we can rewrite the right-hand side of (2.2) as a power series of z 1 , . . . , z n with coefficients g j (τ ) whose q-expansion has no q n -term with n < n 0 . Hence, by Proposition 4.4, D k,p (φ| k,B γ), which is equal to D k,p acting on this power series, has no q n -term with n < n 0 in its q-expansion. This proves the assertion on J ! k,L (G, ρ), while that on J weak k,L (G, ρ) and J k,L (G, ρ) can be proved in a similar manner. Now we proceed to derive the Fourier development of D k,p (φ) := D k,p (emb B (φ)) for weakly holomorphic Jacobi form φ.
Lemma 4.8. Use notations in Convention 1 and 2. Let φ ∈ J ! k,L (G, ρ) and γ ∈ SL 2 (Z) ⋉ H(L, 1). Suppose the Fourier expansion is B(t, z)) , c γ (n, t) ∈ W, and the Taylor expansion with respect to some ordered R-basis B = (e 1 , . . . , e n ) of V is Then we have (4.28) where t = t 1 e 1 + · · ·+ t n e n and G B is the Gram matrix of B with respect to B.
Note that t 1 , . . . , t n in the inner sum may not be rationals, unless we choose B to be a Z-basis of L.
Recall a generalized Cauchy-Schwarz inequality. For any real bilinear form F : V × V → R (whose Gram matrix with respect to the basis B is M), and v 1 , v 2 ∈ V , we have where |·| in the right-hand side is any norm on V and M = sup v∈V |M v| |v| . Thus where M 1 and M 2 are the Gram matrices of B ′ and B ′ − B with respect to B respectively. The quantity |t|, |t ′ |, Q(t) are of the same order of magnitude when t ranging over V . So according to the paragraph following Proposition 2.7, there exists some C > 0 such that |e (B ′ (t ′ , z ′ ) − B(t, z ′ ))| ≤ e C √ n for sufficiently large n and arbitrary t satisfying c γ (n, t) = 0. Thus we can choose any τ 1 with ℑτ 1 < ℑτ in (4.29).
Maybe a better formula is To obtain the Fourier development of D k,p (φ), we introduce certain polynomials.
Proof. From the proof of Theorem 4.6, we know that D k,p (φ)| k+s(p) γ = D k,p (φ| k,B γ). The desired formula then follows from this fact, Proposition 4.4 and Lemma 4.8.
Hence, according to Proposition 4.11, the function D 1/2,(2,0) (φ) is, up to a constant factor, (4.32) This function belongs to M ! 5/2 (Γ 0 (3), χ| Γ 0 (3) ) by Theorem 4.6. The character χ| Γ 0 (3) can be described by its action on generators of Γ 0 (3) as follows: We can collect the operators in Definition 4.2 together to form the following map: where P is any nomempty subset of the vectors in Z n with s(p) ≥ 0. It is clear a C-linear map. A natural question is, for which P this map is an isomorphism? Since we are mainly concerned with weakly holomorphic Jacobi forms of lattice index, we may restrict the domain to M B, 0 k,B (H, ρ) (a special case of (3.16), not to be confused with (3.19)). (1) For any p ∈ Z n ≥0 , we have (2) For any p ∈ Z n ≥0 , we have where C 1 and C 2 (depending on k, s(p) and s(Λ)) are given by The reason why we require k = 0, −1, −2, . . . in Proposition 4.15 is that we need C 1 and C 2 to be always nonzero. We emphasize that an empty product is defined to be 1. When needed, we write C i (p, Λ) instead of C i to indicate their dependence on p and Λ for i = 1, 2.
We now prove Proposition 4.15.
Proof. It is obviously the map is C-linear. To prove the injectiveness, we compute the kernel, which turns out to be zero by the "(1) =⇒ (2)" part of the above lemma. It remains to prove the surjectiveness. Let M O k+s(p) (H, ρ) be arbitrary, and put g ′ p = 0 if p / ∈ Z n ≥0 . Write g p = 4 −λp (λ p !) −1 · g ′ p , and define a formal series j∈Z n h j T j ∈ j≥0 O(H, W)T j by the equation in statement (2) of the above lemma. We shall prove two things: one is j∈Z n h j T j ∈ M B, 0 k,B (H, ρ), the other is D k,p ( j∈Z n h j T j ) = g ′ p , from which the desired surjectiveness follows. The latter assertion is a direct consequence of the above lemma. To prove the former one, let γ = (( a b c d ) , ε) ∈ H be arbitrary. According to Definition 3.12 and Proposition 3.4, we need to prove (4.34) for any p ∈ Z n ≥0 . For this purpose, we shall use the following formula: . This is obtained by using the fact g p ∈ M O k+s(p) (H, ρ) and the statement (1) in Lemma 4.16. Now we use induction on s(p). For the base case s(p) = 0 or 1, the function h p is just g p , so (4.34) holds. For the induction step, let p ∈ Z n ≥0 be arbitrary, and assume that (4.34) has been proved for any p ′ ∈ Z n ≥0 with s(p ′ ) < s(p). Inserting the induction hypothesis into (4.35), and using Lemma 3.5, we obtain, after a tedious simplification, that . Thus, it suffices to show that, for any non-negative integer l and Λ ∈ Z ≥0 n×n with s(Λ) > l, we have or equivalently, (4.36) By (3.6), the desired identity (4.36) is a consequence of the following elementary identity which can be proved by induction on s(Λ) − l. This concludes the induction step, hence the whole proof.

Jacobi theta series and theta decompositions
The main aim of this paper is to find finite subsets P 0 of P = Z n ≥0 , such that the map p∈P 0 D k,p restricted to spaces of weakly holomorphic Jacobi forms (see (4.33) for definition) is still injective. It turns out that another important decomposition techineque, that of theta decomposition, is needed in the proof of this main result (Theorem 6.2, which will be carried out in the next section). In this section, we review some necessary facts of this techineque.
First recall some facts on Jacobi theta series and Weil representations. In this section, unless declare explicitly, we usually assume that L is an even lattice, which could simplify the discussion. Then L ♯ /L can be equipped with a finite quadratic module structure. See [Str13,§2]. We can also form the group algebra C[L ♯ /L], which possesses a Hilbert space structure we now explain. For x ∈ L ♯ /L, let δ x denote the element in C[L ♯ /L] such that δ x (y) = 1 if x = y and δ x (y) = 0 otherwise. Then by the following three formulae: Since SL 2 (Z) has a presentation with generators T = ( 1 1 0 1 ), S = ( 0 −1 1 0 ), and relations S 8 = I, S 2 = ( S T ) 3 , the above three formulae really extend to a representation of SL 2 (Z) ⋉ H(L, 1). Using these three formulae, one can verify that ρ L is a unitary representation, which means ρ L (γ) preserves the inner product on C[L ♯ /L] for any γ ∈ SL 2 (Z) ⋉ H(L, 1). One can also verify that the kernel is of finite index. Note that ρ L , when restricted to SL 2 (Z), is the well-known Weil representation. Strömberg [Str13] has worked out an explicit formula for any ρ L (γ).
Occasionally, we shall use its dual representation (when dealing with theta decompositions) where C[L ♯ /L] * is the dual space (all linear functionals), and ρ L (γ) × means the operator adjoint of ρ L (γ) that maps f ∈ C[L ♯ /L] * to f • ρ L (γ). We remark that the matrix of ρ * L (γ) with respect to the dual basis is the transposed inverse of the matrix of ρ L (γ).
The Jacobi theta series of lattice index is defined by as a scalar-valued function on H × V, where t ∈ L ♯ /L. Note that L can also be an odd lattice. The reader may verify that the above series converges normally, hence uniformly and absolutely on any compact subsets of H × V. Put (5.40) which is a map from H × V to C[L ♯ /L]. It is known that Theorem 5.1. The map ϑ L belongs to J n/2,L (SL 2 (Z), ρ L ).
Second, recall the concept of the tensor product of two group representations. Let ρ 1 and ρ 2 be two group representations on the same abstract group G with representation C-spaces W 1 and W 2 respectively. Then the tensor product ρ 1 ⊗ ρ 2 , is the representation G → GL(W 1 ⊗ C W 2 ) defined by ρ 1 ⊗ ρ 2 (g)(w 1 ⊗ w 2 ) = ρ 1 (g)(w 1 ) ⊗ ρ 2 (g)(w 2 ). Now we review the theory of theta decompositions. For the classical theory, see [EZ85,  for any coprime integers a, b and v ∈ L. Then for any γ ∈ SL 2 (Z) and where h γ t is the holomorphic function on H defined by (5.46) h γ t (τ ) = n c γ (n + Q(t), t)q n , and the quantity c γ (n, t) refers to c(n, t) in (2.2). Moreover, if L is even, then the map that sends φ = n,t c(n, t)q n e (B(t, z)) to In addition, when restricted, this map gives embeddings from J k,L (G, ρ) and J cusp k,L (G, ρ) into M k− 1 2 n (G, ρ⊗ρ * L | G ) and S k− 1 2 n (G, ρ⊗ρ * L | G ) respectively. Proof. Fix φ and γ; we proceed to prove (5.45). A crucial fact is for any (n, t) ∈ R × V , v ∈ L and γ ∈ SL 2 (Z). See the paragraph immediately after Proposition 2.7. Now by (5.44), we have c γ (n + Q(v)+B(t, v), t+v) = c γ (n, t). Proposition 2.5 tells us that c γ (n, t) = 0 implies t ∈ 1 N 1 N 2 L ♯ . But under the assumption (5.44) (put a = 0 and b = 1), we actually have c γ (n, t) = 0 =⇒ t ∈ L ♯ . Taking into account these two facts, and using normal convergence we can rewrite (2.2) as from which (5.45) follows. Also, from the above deduction, we see that (5.46) and (5.47) are well-defined, that is, they are independent of the choice of the representative of t ∈ L ♯ /L, and converge normally, hence are holomorphic on H. Now assume that L is even, and we proceed to prove that the function h given by (5.47) belongs to M ! k− 1 2 n (G, ρ⊗ρ * L | G ). Note that (5.47) can be rewritten as h(τ ) = t∈L ♯ /L h t (τ ) ⊗ δ * t , where h t = h I t with I = ( 1 0 0 1 ). So we first investigate transformation equations of h t . The notation σ t,t ′ is defined by Use this notation, Theorem 5.1 is equivalent to H(L, 1), then by the transofmation laws of φ (see Definition 2.3), and the linear independence 4 of the family ϑ L,t , t ∈ L ♯ /L, we have which is equivalent to We have obtained transformation equations of h t . Next we derive transformation equations of h from those of h t . By the definition of dual representations, we have , from which the transformation laws of h follow. To conclude that h ∈ M ! k− 1 2 n (G, ρ ⊗ ρ * L | G ), it remains to show that, for any γ = (( a b c d ) , ε) ∈ SL 2 (Z), the q-expansion of h| k− 1 2 n γ has at most finitely many terms of negative powers of q. Comparing (5.45) and (5.49) and using the linear independence of the functions z → ϑ L,t (τ, z), t ∈ L ♯ /L, we obtain that The desired property follows from this formula, (5.46) and the conditions c γ (n, t) satisfy (see Definition 2.6). We have shown that the map that sends φ to h is a well-defined map . It is immediately that this map is a C-linear embedding. From the formula (5.51), it is not hard to see that this map gives embedding from J k,L (G, ρ) to M k− 1 2 n (G, ρ ⊗ ρ * L | G ), and from J cusp k,L (G, ρ) to S k− 1 2 n (G, ρ ⊗ ρ * L | G ). This concludes the proof. 4 In fact, for any fixed τ ∈ H, the functions z → ϑ L,t (τ, z), t ∈ L ♯ /L are C-linear independent. This follows from the uniqueness of coefficients of Fourier series.
Remark 5.3. Suppose L is even. Applying the operator | k− 1 2 n ( 1 0 0 1 ) to (5.47), and using (5.50), we obtain that H(L, 1). This is equivalent to So if the set {h t (τ ) : τ ∈ H, t ∈ L ♯ /L} generates W, then ρ must satisfy Without loss of generality, for even lattices, we mostly consider ρ satisfying this condition. In fact, under this condition, by reversing the deduction in the above proof, one can show that the map from J ! k,L (G, ρ) to M ! k− 1 2 n (G, ρ ⊗ ρ * L | G ) in Theorem 5.2 is really an isomorphism. So are its restrictions to J k,L (G, ρ) and to J cusp k,L (G, ρ). Nevertheless, we do not need the surjectiveness below.
In the remaining of this section, we describe the relationship between Taylor expansions and theta decompositions. For results in the classical setting, see [BFOR17,Proposition 2.40]. Although we could obtain a formula relating modified Taylor coefficients D k,p (φ) to theta coefficients (5.46) concerning Rankin-Cohen brackets, we only present a formula involving non-modified Taylor coefficients, for this is precisely what we need to prove the main result (Theorem 6.2) of this paper.
For a R-basis B of V , let (z 1 , z 2 , . . . , z n ) denote the corresponding coordinate map on V. Let p ∈ Z n ≥0 . By ∂ p B , or simply ∂ p , we mean the differential operator ∂ s(p) Equivalently, for any t ∈ L ♯ /L, we have Proof. Immediately from definitions, that is, (5.39) and (5.40).
Remark 5.6. Note that (v B · G B ) p can be rewritten as n l=1 B(v, e l ) p l where p l is the l-th component of p and B = (e 1 , . . . , e n ).
Proposition 5.7. Use notations in Convention 1 and 2. The lattice L may be even or odd. Suppose that ρ satisfies (5.53). Let φ = n,t c(n, t)q n e (B(t, z)) ∈ J ! k,L (G, ρ) and put h t (τ ) = n c(n+Q(t), t)q n for t ∈ L ♯ /L. Let B = (e 1 , . . . , e n ) be a R-basis of V and G B be the Gram matrix of B with respect to B. Suppose the Taylor expansion of φ with respect to B is φ(τ, z 1 e 1 + . . . z n e n ) = j∈Z n ≥0 g j (τ )z j 1 1 . . . z jn n .
Then we have Proof. This follows from (5.45) with γ = ( 1 0 0 1 ) and Lemma 5.5. Corollary 5.8. Use notations and assumptions above. Let P 0 = {p 1 , . . . , p d } be a finite subset of Z n ≥0 with d = |L ♯ /L| elements. Assume L ♯ /L = {t 1 , . . . , t d }. If the function is not identically zero for τ ∈ H, then . By Lemma 5.5 and Proposition 5.7 we have Choose some τ 0 such that the function (5.54) assumes a non-zero value. By continuity (5.54) assumes non-zero values on a neighbourhood U of τ 0 . Thus, inverting (5.55) gives that h t j (τ ) = 0 with τ ∈ U, 1 ≤ j ≤ d. By analytic continuation h t j are all identically zero, and hence by theta expansion (actually by Remark 5.4) φ = 0.
Definition 5.9. For any integral lattice L of determinant d, any Rbasis B of V , and any finite subset P 0 ⊂ Z n ≥0 of cardinality d, we denote (5.54) by F L,P 0 (τ ), or more precisely F B L,P 0 (τ ). Note that for fixed B, F L,P 0 (τ ) actually depends on orderings P 0 and L ♯ /L possess. But different choices of orderings only possibly lead to functions differ by a factor −1.
Since the function F L,P 0 (τ ) is holomorphic, for any specific L and P 0 , we can check the desired condition by writing out the q-expansion of F L,P 0 (τ ). Computer algebra systems are suitable for such task.
Example 5.10. Suppose L is even and ρ satisfies (5.52). If it is also unimodular, that is, |L ♯ /L| = 1, then the structure of J ! k,L (G, ρ) is in some sense the simplest. For instance, consider the E 8 lattice, i.e., the module Z 8 equipped with the bilinear form whose Gram matrix with respect to the standard basis of Z 8 is the Cartan matrix of the E 8 root system. (See [TY05,p. 272] for this matrix.) We identify See also [Kri96,Corollary 3]. On the other hand, according to Theorem 4.6 (with B the standard basis), we have a C-linear map We assert that, the above map D k,0 , is an embedding. This follows immediately from Corollary 5.8. In fact, this assertion holds for any unimodular lattice, whether it is even or old.
For more concrete examples, see Section 7.

The main theorem and its corollaries
Throughout this section, we adopt Convention 1 and 2. Recall the definition of emb B before Proposition 3.6. For φ ∈ J ! k,L (G, ρ), if the basis B is implicitly known or fixed, we oftern write D k,p (φ), instead of D k,p (emb B φ), as in the proof of Theorem 4.6.
We begin with the following fact, which is a direct consequence of Proposition 4.15.
Lemma 6.1. Assume that k = 0, −1, −2, · · · . Fix a R-basis B of V . Then the map Proof. The space in which the image of φ lies can be determined by Theorem 4.6. The injectiveness follows from Proposition 4.15.
To state our main theorem, some more notations are needed. We define a partial ordering on Z n ≥0 as follows. Let p 1 and p 2 be vectors in Z n ≥0 . By p 1 p 2 , we mean p 2 −p 1 ∈ Z n ≥0 and s(p 1 ) ≡ s(p 2 ) mod 2. Let P be a subset of Z n ≥0 . We put P = {p ∈ Z n ≥0 : p p ′ for some p ′ ∈ P}. If P is a singleton, that is P = {p 0 }, then we also write p 0 instead of P. The motivation for concerning this partial ordering is that the p-th Taylor coefficient of a Jacobi form φ is precisely determined by D k,p ′ φ with p ′ ∈ p. See Lemma 4.16.
We can now state and prove our main theorem. We emphasize that D k,p denotes D B k,B,p • emb B (see Definition 4.2 and the paragraph before Proposition 3.6), ϑ L,t denotes Jacobi theta series (5.39), and ∂ p is defined in the paragraph preceding Lemma 5.5.
Proof. The fact that the map considered is C-linear and that it maps J weak k,L (G, ρ) to p∈ P 0 M k+s(p) (G, ρ| G ), and J k,L (G, ρ) to M k (G, ρ| G ) × 0 =p∈ P 0 S k+s(p) (G, ρ| G ) follow from 5 Lemma 6.1. So it remains to show the injectiveness, for which we compute the kernel. Let φ ∈ J ! k,L (G, ρ) such that D k,p ′ φ = 0 for any p ′ ∈ P 0 . By Lemma 4.16(2), the pth Taylor coefficient of z → φ(τ, z) (under the basis B) is a linear combination of (6.56) Hence, if p ∈ P 0 , then the p-th Taylor coefficient is zero since p ⊆ P 0 . It follows immediately from this fact and Corollary 5.8 that φ = 0. This proves the injectiveness, hence concludes the proof.
Remark 6.3. To apply this theorem, it is essential to obtain a set P 0 first. Note that the choice of P 0 depends only on the lattice L = (L, B), not on the weight k, the group G or the representation ρ. In the remaining, we make no effort to prove the existence of P 0 for all lattices theoretically, but only try to find out P 0 for some specific lattices computationally.
Remark 6.4. In certain circumstances, we may drop or loosen the restriction k = 0, −1, −2, · · · , i.e., k may be some non-positive integer. To see this, note it is in the step showing the p-th Taylor coefficient of z → φ(τ, z) is a linear combination of (6.56) that we use the fact k = 0, −1, −2, · · · , for Lemma 4.16 requires this. Hence, for fixed P 0 , one may analyse expressions C 1 and C 2 in Lemma 4.16(2) to seek those k's such that these two expressions are both nonzero for any p ∈ P 0 . For such k, Theorem 6.2 still holds. In the remaining, we always assume that k = 0, −1, −2, · · · for the sake of simplicity.
Concrete examples will be given in the next section. The rest of this section is devoted to some of immediate corollaries concerning lattices of small determinants. All these corollaries assume the same notations and conditions as Theorem 6.2.  = (0, . . . , 0, 2). Then the following three maps are all C-linear embeddings.
Proof. In this case, P 0 = {0, p}. So P 0 = P 0 . By Theorem 6.2, we shall verify that where (v 1 , . . . , v n ) and (α 1 , . . . , α n ) are coordinate vectors of v and α with respect to B respectively, and g ij is the (i, j)-entry of the Gram matrix of B with respect to B. Set m = min{Q(α + v) : v ∈ L}. Then the q m -term of the left-hand side of the desired inequality is non-zero, since (α 1 + w 1 )g n1 + . . . (α n + w n )g nn = B(α + w, e n ) = 0. This concludes the proof.
Hence by Proposition 4.11, we have D k,p φ = 2 · (2πi) 2 n t c(n, t)(−g nn n + k(g 1n t 1 + · · · + g nn t n ) 2 ) q n , where φ = n,t c(n, t)q n e (B(t, z)) and (t 1 , . . . , t n ) is the coordinate vector of t with respect to B. This formula is general for all lattices, not only for those discussed in the above corollary. The expression (4.32) is essentially a special case of this formula.
Proof. In this case, P 0 = {0, p 1 , p 2 }. So P 0 = P 0 . By Theorem 6.2, we shall verify that the determinant of the following matrix is not identically zero: All above summations are over v ∈ L. Put m = min v∈L Q(α + v) = min v∈L Q(2α + v) and let a ij be the (i, j)-entry of this matrix. By positive definiteness we have m > 0. By a change of variables we see that a 12 = a 13 , a 22 = −a 23 and a 32 = a 33 , so the determinant is 2a 22 (a 11 a 32 − a 12 a 31 ). By assumption, a 22 = 0, and by the existence of w, the q m -term of a 11 a 32 − a 12 a 31 is nonzero. Hence the determinant is nonzero, which concludes the proof.
Remark 6.8. As in Remark 6.6, we write down the explicit formula of D k,p 1 φ with p 1 = (0, . . . , 0, 1). By Definition 4.10, we have P k,p 1 ,G B (X 0 , X 1 , . . . , X n ) = 2(g 1n X 1 + · · · + g nn X n ), where g ij is the (i, j)-entry of the Gram matrix G B of B with respect to B. Hence by Proposition 4.11, we have where φ = n,t c(n, t)q n e (B(t, z)) and (t 1 , . . . , t n ) is the coordinate vector of t with respect to B.
For L with large determinant, it is computationally inefficient to evaluate F L,P 0 directly. We provide a useful criterion for F L,P 0 to be nonzero below, which works in some situations. Proposition 6.9. Use notations in Theorem 6.2. Suppose n ≥ 2, p 1 = 0, and t 1 = 0. Put for i = 2, 3, . . . , d. For p i ∈ P 0 , the symbol p i,l denotes its l-th component. Suppose the ordered basis B equals (e 1 , . . . , e n ). If the is non-singular, then F L,P 0 is not identically zero.
Proof. Put m j = min v∈t j +L Q(v) for j = 2, . . . , d, and Treat a ij as its power series expansion with respect to q = e (τ ) (with possibly fractional powers). By Lemma 5.5, v∈S j n l=1 B(v, e l ) p i,l is the coefficient of q m j -term of a ij for i, j ≥ 2. Since the constant term of a 11 is 1 and that of a i1 is 0 for i ≥ 2, the determinant of (6.57) is equal to the coefficient of the q 2≤j≤d m j -term of det(a ij ) 1≤i,j≤d . Hence from the non-singularity of (6.57) F L,P 0 is not identically zero.
What can we use an embedding to do? One immediate application is to find linear relations among functions in the domain of the embedding. Such a relation holds, if and only if the corresponding relation in the codomain holds. The codomian of the embedding in Theorem 6.2 is roughly a product (direct sum) of spaces of modular forms, in which we have standard algorithm to find linear relations, combining basic linear algebra and some theory of ordinary modular forms, at least in the scalar-valued case. In the next section, we will do this for certain Jacobi theta series of lattice index.

Application: Linear relations among theta series
Functions studied in this section are the followings.
Note that we omit the information L in the notation.
For fixed L, there are totally N n such theta series. But there is something subtle in this definition: the function θ α,β actually depends on the representatives we choose for α and β. Different choice of representatives leads to functions differ by a constant factor. So this ambiguity may not cause trouble. One of the aims of this section is to find linear relations among {θ N α,β : α ∈ L ♯ /L, β ∈ 1 N L/L ♯ }, for some specific lattices L. Note that θ N α,β is independent of the choice of representatives of α and β. Furthermore, if N ′ is a positive divisor of N and β satisfies N ′ β = 0 + L ♯ , then θ N ′ α,β is also independent of the choice of representatives.
To apply Theorem 6.2 to functions in Definition 7.1 (including those in the above two examples), we need to know their transformation laws. Recall that (7.62) Lemma 7.4. Suppose L is even, N is the level of L, and t ∈ L ♯ /L.
Remark 7.5. One can work out an explicit expression for the generalized Gauss sum g(b, d; t). See [CS17, §14.3] for instance. But we do not need this. Moreover, the quantity g(b, d; t) actually depends on the lattice, so we write g L (b, d; t) when needed.
We give a sketch of proof below for the readers' convenience. See also [CS17,§14.3] for details of this proof.
Proof. The case c = 0 can be proved directly. So assume that c = 0 below. We first prove a more general formula: where γ = ( a b c d ) ∈ SL 2 (Z) (not only Γ 0 (N)), and (7.65) Using (5.42) and the identity aτ +b cτ +d = a c − 1 c(cτ +d) , we have , where ε(c, d) = −1 if c, d are both negative, and ε(c, d) = 1 otherwise. Hence ε(c, d)) . Now (7.64) follows from applying (7.66) with ( a b c d ) replaced by −b a −d c , and then using (5.42) in the above formula.
Then we prove the original formula from (7.64). Consider the group homomorphism Since ( a b c d ) ∈ Γ 0 (N) by assumption, we have cL ♯ ⊆ L, so the above homomorphism is an isomorphism. Applying a change of variable corresponding to this isomorphism in the inner sum of the right-hand side of (7.65), we obtain that The lemma follows from this and (7.64).
The requirement of Fourier expansions of Jacobi forms in Definition 2.6 now follows from this identity and Theorem 5.1. It remains to show the transformation laws of θ α,β . To do this, assume that ( a b c d ) ∈ Γ 1 (N).
Remark 7.7. When d = 0, we must have N = 1. So L is an even unimodular lattice. The value of χ α,β at such (( a b c d ) , ε, [v, w], ξ) can also be evaluated by (7.68), while we do not need this. Proof. This follows immediately from Proposition 7.6 and Proposition 2.10.
Some cautions must be mentioned. Here we have two bilinear forms B and N · B, so concepts depending on the bilinear form must be carefully treated. For instance, the slash operators | n/2,B and | N n/2,N ·B are different. For another instance, the Gauss sum occurring in the above corollary is We also state a slight generalization of Corollary 7.8 below, whose proof is nearly the same as that of Corollary 7.8.
Proposition 7.9. Let N ′ be a positive divisor of the level N. Then the functions θ N ′ α,β with α, β satisfying Q(α) ∈ 1 N ′ Z and β ∈ (L ♯ + 1 N ′ L)/L ♯ all lie in the same space Now we apply Theorem 6.2 to functions θ N ′ α,β , to explore linear relations among them. We shall first give a general algorithm (Theorem 7.10), the main result of this section, and then present four examples concerned with low-dimensional root lattices, and one concerned with binary quadratic form.
One can verify immediately that the coefficient of the q n -term in the above sum is equal to (7.71), which is denoted by Θ α,β (p, n) from now on. Therefore, the relation (7.72) holds, if and only if (7.73) α,β a α,β Θ α,β (p, n) = 0 for any p ∈ P 0 and any n ∈ Q. It remains to show that, we can replace the condition "n ∈ Q" with "n belonging to the numbers given by (7.70)" in the above statement. On the one hand, since we have c α,β (n, t) = 0 unless n ∈ Z ≥0 , and so (7.73) holds automatically for n / ∈ Z ≥0 . On the other hand, it follows from the fact In principle, for any specific even integral lattice, one can find out a maximal linearly independent set of functions among θ N α,β 's, and all linear relations among them, by analysing the corresponding vectors Θ α,β using a computer algebra system, provided that we can first find a P 0 such that F L N ,P 0 = 0. We use SageMath 9.3 [Sag21] here. Now consider specific lattices. First recall the concept of root lattices. The ambient real space is V = (V, B), as above. Let L be an integral lattice in V . We say L is a root lattice in V , if there is a Z-basis of L containing only vectors v such that Q(v) = 1. See [Ebe13,§1.4] for more details, or [TY05, Appendix of §18] for the classification and a full description of all irreducible root systems.
The set P 0 now has cardinality 94, and the set I in Theorem 7.10 has cardinality 258. There are totally 94 polynomials P 4,p,2·G B 's to calculate first. By the SageMath program, we work out the vectors Θ α,β (whose components in this case, are all integers) and find out a maximal linearly independent set and all linear relations, as follows.
There are nine theta series θ α,β associated with A 2 , where α is chosen from (7.76), and β is chosen from (7.77). They are all defined on H×C 2 , and the third power of them all lie in the space J 3,A 2 3 (Γ 1 (3), χ 3 ), where the lattice A 2 3 is (Z 2 , 3 · B). See Definition 7.1 and Corollary 7.8 for details. The linear relations among the third power of these nine functions are known by D. Schultz. In a systematic treatment of cubic theta functions [Sch13], he defined nine theta functions, Θ ij (u | τ ) in his notation, in Section 2.3. One may check that, Schultz's functions are essentially functions θ α,β associated with A 2 defined here, up to some trivial changes of variables. So the following theorem should be attributed to Schultz. We only provide an alternate proof here, based on Theorem 7.10 and the aid of any computer algebra system.
To apply Theorem 7.10 to functions θ 2 α,β , we first find a set P 0 ⊂ Z 3
Hence F A 3 2 ,P 0 = 0, and Theorem 7.10 is applicable for this P 0 .
The set P 0 now has cardinality 43, and the set I in Theorem 7.10 has cardinality 635. There are totally 43 polynomials P 3,p,2·G B 's to calculate first. By the SageMath program, we work out the vectors Θ α,β (whose components in this case, are all integers) and find out a maximal linearly independent set and all linear relations, as follows.

Miscellaneous observations and open questions
8.1. The case of odd lattices. Our main theorem, Theorem 6.2, is valid for both even lattices and odd lattices. In fact, we have made much effort to state many propositions general for both kinds of lattices. The first place where even and odd lattices behave differently is in Theorem 5.1. And then in Theorem 5.2, the assertion for even lattices is stronger than that for odd lattices, while the weak version (that is, Remark 5.4) for both kinds of lattices suffices in the proof of the main theorem.
However, the whole Section 7, in particular Theorem 7.10, concerns only even lattices. It is for the reason of simplicity. Actually, there is a version of Theorem 7.10 that works for both cases. Here we briefly describe what this general version looks like.
First, we generalize Theorem 5.1 to allow odd lattices. Let Γ θ be the subgroup 7 of SL 2 (Z) generated by ( 1 2 0 1 ) and ( 0 −1 1 0 ). Then one can prove that ϑ L ∈ J n/2,L (Γ θ , ρ L ), where L is any integral lattice in V = (V, B) and ρ L is the group representation of Γ θ ⋉ H(L, 1) defined by the same formulae used to define (5.37), except for the first formula, which should be replaced by ρ L ( 1 2 0 1 )δ x = e (2Q(x)) δ x . One may prove that these three formulae indeed can be extended to a representation of Γ θ ⋉ H(L, 1) (by transformation laws for Jacobi theta series or by firstly finding a presentation of Γ θ ). More general transformation formulae, (7.66), still hold for odd lattices, provided that ( a b c d ) ∈ Γ θ and c = 0. The proof is almost the same as that for even lattices. In this proof, factors like e (acQ(v)) and e (bdQ(v)) occur (v ∈ L). It is because ( a b c d ) ∈ Γ θ that such factors all equal 1. Another version of general transformation formulae, (7.64), again holds for odd lattices, provided that ( a b c d ) ∈ Γ θ and d = 0. (c = 0 is permitted.) Next, consider Definition 7.1. It also works for odd lattices, but perhaps the least positive integer N such that N ·Q(v) ∈ Z for all v ∈ L ♯ is not commonly called "level" in the context of odd lattices, as we shall do. Lemma 7.4 can be immediately generalized to cover the case of odd lattices -just replace the condition ( a b c d ) ∈ Γ 0 (N) by ( a b c d ) ∈ Γ 0 (N) ∩ Γ θ and leave the remaining part unchanged. Proposition 7.6, Corollary 7.8 and Proposition 7.9 are based on Lemma 7.4, so they can also be generalized to include the odd-lattice case by replacing the group, say Γ 1 (N), with Γ 1 (N) ∩ Γ θ . But there is some subtle difference in values of χ α,β in Proposition 7.6. For odd lattice L, the value of χ α,β which is not totally same as that in Proposition 7.6. Moreover, we state the odd-lattice version of Proposition 7.9 explicitly.
Proposition 8.1. Let L be odd, N ′ be a positive divisor of the "level" N such that N/N ′ is even. Then the functions θ N ′ α,β with α, β satisfying α ∈ L ♯ /L, Q(α) ∈ 1 2N ′ Z and β ∈ (L ♯ + 1 N ′ L)/L ♯ all lie in the same space J N ′ n 2 ,L N ′ (Γ 1 (N) ∩ Γ θ , χ N ′ ), where χ N ′ is the linear character on Γ 1 (N) ∩ Γ θ ⋉ H(L N ′ , 1) that maps (( a b c d ) , ε, [v, w], ξ) (in this case d must be nonzero) to Finally, we are now able to generalize Theorem 7.10 to odd lattices. We shall make following changes: (1) The lattice L is now assumed to be an odd lattice in V = (V, B).
(2) Besides the condition (α, β) ∈ L ♯ /L × (L ♯ + 1 N ′ L)/L ♯ , we need another assumption 2 | N/N ′ . We shall also replace the condition Q(α) ∈ 1 N ′ Z by Q(α) ∈ 1 2N ′ Z. 8.2. The image of the embedding in the main theorem. In Theorem 6.2, we establish embeddings from certain space of Jacobi forms to some direct product of finitely many spaces of modular forms. It is natural and interesting to describe images of such embeddings. In the context of classical Jacobi forms (integral-index, scalar-valued, integralweight, and trivial character), Eichler and Zagier have found that the space of weak Jacobi forms is isomorphic to a direct product of finitely many spaces of modular forms ([EZ85, Theorem 9.2]). We wish to investigate the general case: Open question. Use notations and assumptions in Theorem 6.2. We invite interested readers to describe images of spaces J ! k,L (G, ρ), J weak k,L (G, ρ), J k,L (G, ρ) and J cusp k,L (G, ρ) under the map in that theorem. In particular, we ask that, is the space J weak k,L (G, ρ) isomorphic to some product of finitely many spaces of modular forms?
There is another interesting question concerned with these embeddings: Open question. Use notations and assumptions in Theorem 6.2. What is the set P 0 with the least number of elements, such that the map is injective? How can we describe such set theoretically, not computationally?
In this direction, for the integral-index setting, there are some good works sharpening the original reslut of Eichler and Zagier. For instance, consult [Kra86], [AB99], [RS13], [DR15] and [DP17]. We wish that, their ideas could also be used to deal with lattice-index case. for p in P0: 9 re |= set(closure_preceq(p)) 10 return re The parameter p of the function closure_preceq represents a vector p ∈ Z n ≥0 , and a call of closure_preceq(p) returns a Python list representing p. The parameter P0 of the function closure_preceq_set represents a finite subset P 0 of Z n ≥0 , and a call of closure_preceq_set(P0) returns a Python set representing P 0 .