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Stable Schottky-Jacobi Forms

In this article, we prove that there do not exist stable Schottky-Jacobi forms for the universal Jacobian locus and also prove that there exist non-trivial stable Schottky-Jacobi forms for the universal hyperelliptic locus.


Introduction
For a positive integer g, we let H g = τ ∈ C (g,g) | τ = t τ, Im τ > 0 be the Siegel upper half plane of degree g and let Sp(2g, R) = {M ∈ R (2g,2g) | t M J g M = J g } be the symplectic group of degree g, where F (k,l) denotes the set of all k × l matrices with entries in a commutative ring F for two positive integers k and l, t M denotes the transposed matrix of a matrix M and J g = 0 I g −I g 0 .
Then Sp(2g, R) acts on H g transitively by where M = A B C D ∈ Sp(2g, R) and Ω ∈ H n . Let Γ g = Sp(2g, Z) = A B C D ∈ Sp(2g, R) A, B, C, D integral be the Siegel modular group of degree g. This group acts on H g properly discontinuously.
Let A g := Γ g \H g be the Siegel modular variety of degree g, that is, the moduli space of g-dimensional principally polarized abelian varieties, and let M g be the the moduli space of projective curves of genus g. Then according to Torelli's theorem, the Jacobi mapping T g : M g −→ A g defined by C −→ J(C) := the Jacobian of C is injective. The Jacobian locus J g := T g (M g ) is a (3g − 3)-dimensional subvariety of A g The Schottky problem is to characterize the Jacobian locus or its closureJ g in A g . At first this problem had been investigated from the analytical point of view : to find explicit equations of J g (orJ g ) in A g defined by Siegel modular forms on H g , for example, polynomials in the theta constant θ ǫ δ (τ, 0) and their derivatives. The first result in this direction was due to Friedrich Schottky [22] who gave the simple and beautiful equation satisfied by the theta constants of Jacobians of dimension 4. Much later the fact that this equation characterizes the Jacobian locus J 4 was proved by J. Igusa [14] (see also [9], [11] and [13]). Past decades there has been some progress on the characterization of Jacobians by some mathematicians. For two positive integers g and h, we consider the Heisenberg group H (g,h) R = (λ, µ; κ) | λ, µ ∈ R (h,g) , κ ∈ R (h,h) , κ + µ t λ symmetric endowed with the following multiplication law . We define the Jacobi group G J of degree g and index h that is the semidirect product of Sp(2g, R) and H and (Ω, Z) ∈ H g × C (h,g) . We note that the Jacobi group G J is not a reductive Lie group and the homogeneous space is not a symmetric space. From now on, for brevity we write H g,h = H g × C (h,g) . The homogeneous space H g,h is called the Siegel-Jacobi space of degree g and index h.
be the Jacobi modular group. Let A g,h := Γ J g \H g,h be the universal abelian variety. Consider the natural projection map π g,h : A g,h −→ A g .

Let
J g,h := π −1 g,h (J g ) be the universal Jacobian locus and let Hyp g,h := π −1 g,h (Hyp g ) be the universal hyperelliptic locus, where Hyp g is the hyperelliptic locus in A g .
Let 2M be a positive definite, even unimodular integral symmetric matrix of degree h. According to Theorem 3.6 in [27], if g + h > 2k + 1 with a nonnegative integer k, the Siegel-Jacobi operator is an isomorphism (see also Theorem 2.2). Using this fact, we define the notion of stable Jacobi forms of weight k and index M. A Jacobi form F ∈ J k,M (Γ g ) is said to be a Schottky-Jacobi form for J g,h (resp. Hyp g,h ) if it vanishes along J g,h (resp. Hyp g,h ). In a natural way, we can define the notion of stable Schottky-Jacobi forms of index M. For precise definitions, we refer to Definition 2.3 and Definition 4.2.
The aim of this paper is to prove the non-existence of stable Schottky-Jacobi forms for the universal Jacobian locus and also to prove that there exist non-trivial stable Schottky-Jacobi forms for the universal hyperelliptic locus.
This article is organized as follows. In Section 2, we review some properties of the Siegel-Jacobi operator and the notion of stable Jacobi forms introduced by J.-H. Yang [30]. In Section 3, we review the notion of stable Schottky-Siegel forms and the works that were done recently by G. Codogni and N. I. Shepherd-Barron [3,4]. In Section 4, we introduce the notion of stable Schottky-Jacobi forms and prove the following two theorems.  In the final section, we make some comments and present several questions.

Notations:
We denote by Q, R and C the field of rational numbers, the field of real numbers and the field of complex numbers respectively. We denote by Z and Z + the ring of integers and the set of all positive integers respectively. R + denotes the set of all positive real numbers. Z + and R + denote the set of all nonnegative integers and the set of all nonnegative real numbers respectively. The symbol ":=" means that the expression on the right is the definition of that on the left. For two positive integers k and l, F (k,l) denotes the set of all k × l matrices with entries in a commutative ring F . For a square matrix A ∈ F (k,k) of degree k, σ(A) denotes the trace of A. For any M ∈ F (k,l) , t M denotes the transpose of a matrix M . I n denotes the identity matrix of degree n. We put i = √ −1.

Stable Jacobi Forms
For a non-negative integer k, we denote by [Γ g , k] the vector space of all Siegel modular forms of weight k. The Siegel Φ-operator is an important linear map defined by H. Maass [18] proved that if k is even and k > 2g, then Φ g is surjective. E. Freitag [7] proved that if g > 2k, then Φ g is injective. Using the theory of singular modular forms developed by Freitag [8,10], he showed the following: is called a stable modular form of weight k if it satisfies the following conditions (SM1) and (SM2): Let ρ be a rational representation of GL(g, C) on a finite dimensional complex vector for G J on H g,h is given as follows : and (τ, z) ∈ H g,h . We refer to [29] for a geometrical construction of J ρ,M .
Definition 2.2. Let ρ and M be as above. Let . A Jacobi form of index M with respect to ρ on a subgroup Γ of Γ g of finite index is a holomorphic function f ∈ C ∞ (H g,h , V ρ ) satisfying the following conditions (A) and (B): has a Fourier expansion of the following form : If g ≥ 2, the condition (B) is superfluous by Köcher principle ( cf. [33] Lemma 1.6). We denote by J ρ,M (Γ) the vector space of all Jacobi forms of index M with respect to ρ on Γ. Ziegler ( cf. [33] Theorem 1.8 or [6] Theorem 1.1 ) proved that the vector space J ρ,M (Γ) is finite dimensional. In the special case ρ(A) = (det(A)) k with A ∈ GL(g, C) and a fixed k ∈ Z, we write J k,M (Γ) instead of J ρ,M (Γ) and call k the weight of the corresponding Jacobi forms. For more results about Jacobi forms with g > 1 and h > 1, we refer to [26,27,28,29,30,31,32] and [33]. Jacobi forms play an important role in lifting elliptic cusp forms to Siegel cusp forms of degree 2g (cf. [16,17]). Now we consider the special case ρ = det k with k ∈ Z + . We define the Siegel-Jacobi operator . We observe that the above limit exists and Ψ g,M is a well-defined linear map (cf. [33]). J.-H. Yang [27] proved the following theorems.
Theorem 2.1. Let 2M be a positive even unimodular symmetric integral matrix of degree h and let k be an even nonnegative integer. If g + h > 2k, then the Siegel-Jacobi operator Ψ g,M is injective.
Theorem 2.2. Let 2M be as above in Theorem 2.1 and let k be an even nonnegative integer. If g + h > 2k + 1, then the Siegel-Jacobi operator Ψ g,M is an isomorphism.
Let 2M be as above in Theorem 2.1. Yang proved that if k is an even nonnegative integer and g + rank(M) > 2k, then any non-zero Jacobi form in J k,M (Γ g ) is singular (cf. [ (SJ2) Ψ g,M F g = F g−1 for all g ≥ 1.
Remark 2.3. The concept of a stable Jacobi forms was introduced by Yang [30].
Example. Let S be a positive even unimodular symmetric integral matrix of degree 2k and let c ∈ Z (2k,h) be an integral matrix. We define the theta series ϑ It is easily seen that ϑ is a stable Jacobi form of weight k and index M.

Stable Schottky-Siegel Forms
Let A Sat g be the Satake compactification of the Siegel modular variety A g (cf. [21]).
W. Baily [1] proved that A Sat g is a normal projective variety in which A g is Zariski open. In particular, we have a closed embedding The collection (A Sat g ) g≥0 and the above embeddings (ι g ) g≥0 define the projective limit which is called the stable Satake compactification. Let L g be the determinant line bundle of the Hodge bundle over A g . The we have the isomorphism Let J Sat g (resp. Hyp Sat g ) be the closure of J g (resp. Hyp g ) inside A Sat g . We define Q It is well known that θ Q,g (τ ) is a Siegel modular form on H g of weight m 2 . We easily see that Φ g+1 (θ Q,g+1 ) = θ Q,g for all g ≥ 0. Therefore the collection of all theta series associated to (Λ, Q) is a stable modular form of weight m 2 . Freitag [8] proved the following theorem.
Theorem 3.1. The ring of stable modular forms is a polynomial ring in countably many theta series Θ Q = (θ Q,g ) g≥0 associated to irreducible positive even unimodular quadratic forms.

Definition 3.3.
A modular form f ∈ [Γ g , k] is called a Schottky-Siegel form of weight k for J g (resp. Hyp g ) if it vanishes along J g (resp. Hyp g ). A collection (f g ) g≥0 is called a stable Schottky-Siegel form of weight k for the Jacobian locus (resp. the hyperelliptic locus) if (f g ) g≥0 is a stable modular form of weight k and f g vanishes along J g (resp. Hyp g ) for every g ≥ 0.
be the Igusa modular form, that is, the difference of the theta series in genus g associated to the two distinct positive even unimodular quadratic forms E 8 ⊕ E 8 and D + 16 of rank 16. We see that ϕ g (τ ) is a Siegel modular form on H g of weight 8. Since Φ g ϕ g = ϕ g−1 for all g ≥ 1, a collection (ϕ g ) g≥0 is a stable modular form of weight 8. Igusa [14,15] showed that the Schottky-Siegel form discovered by Schottky [22] is an explicit rational multiple of ϕ 4 . In [14], he also showed that the Jacobian locus J 4 is reduced and irreducible, and so cuts out exactly J 4 in A 4 . Indeed, ϕ 4 (τ ) is a degree 16 polynomial in the Thetanullwerte of genus 4. On the other hand, Grushevsky and Salvati Manni [12] showed that the Igusa modular form ϕ 5 of genus 5 cuts out exactly the trigonal locus in J 5 and so does not vanish along J 5 . Thus (ϕ g ) g≥0 is not a stable Schottky-Siegel form.
G. Codogni [3] proved the following. Theorem 3.3. There exist non-trivial stable Schottky-Siegel form for the hyperelliptic locus. Precisely the ideal of stable Schottky-Siegel forms for the hyperelliptic locus is generated by differences of theta series Θ P − Θ Q , where P and Q are positive definite even unimodular quadratic forms of the same rank.
Remark 3.2. Let P and Q be two positive even unimodular quadratic forms of the same rank. We let Θ P := (θ P,g ) g≥0 and Θ Q := (θ Q,g ) g≥0 be two stable modular forms. Codogni [3,Theorem 1.4] showed that the difference of theta series Θ P − Θ Q is a stable Schottky-Siegel form for the hyperelliptic locus when one of the following conditions (1)-(3): (1) rank(P )=rank(Q)=24 and the two quadratic forms have the same number of vectors of norm 2; (2) rank(P )=rank(Q)=32 and the two quadratic forms do not have any vectors of norm 2; (3) rank(P )=rank(Q)=48 and the two quadratic forms do not have any vectors of norm 2 and 4.

Stable Schottky-Jacobi Forms and Proofs of Main Theorems
In this section, we introduce the notion of stable Schottky-Jacobi forms and prove the main theorems.
We let A g,h := Γ J g,h \H g,h be the universal abelian variety and let A Sat g,h := A g,h ∪ A g−1,h ∪ · · · ∪ A 1,h ∪ A 0,h be the Satake compactification of A g,h . We consider the natural projection map   (SSJ1) F g ∈ J k,M (Γ g ) is a Schottky-Jacobi form of weight k and index M for all g ≥ 0.
(SSJ2) Ψ g,M F g = F g−1 for all g ≥ 1. Proof. We first observe that h ≡ 0 (mod 8) (cf. [23]). Assume that there exists a non-trivial stable Schottky-Jacobi form (F g ) g≥0 of weight k and index M for the universal Jacobian locus.
On the other hand, by the assumption that (F g ) g≥0 is a stable Schottky-Jacobi form, we have for all g ≥ 1. Therefore Φ g f g = f g−1 for all g ≥ 1.
Obviously f g vanishes along J g for all g ≥ 0. Thus (f g ) g≥0 is a non-trivial stable Schottky-Siegel form of weight k * . This contradicts the non-existence of a non-trivial stable Schottky-Siegel form for the Jacobian locus (see Theorem 3.2).
Case 2: k is odd.
Then we see easily that a collection ξ = (ξ g ) g≥0 is a non-trivial stable Schottky-Siegel form of weight 2k − h for the Jacobian locus. This contradicts the non-existence of a non-trivial stable Schottky-Siegel form for the Jacobian locus. Hence we complete the proof of the above theorem(=Theorem 1.1).
Theorem 4.2. Let 2M be a positive even unimodular symmetric integral matrix of degree h. Then there exist non-trivial stable Schottky-Jacobi forms of M for the universal hyperelliptic locus.
Proof. According to Theorem 3.3, there exists a non-tivial stable Schottky-Siegel form (f g ) g≥0 of weight k for the hyperelliptic locus. We see from (SO1) in Section 2 that k ≡ 0 (mod 4) and k ∈ Z + . We put ℓ := k + h 2 . Then using the Shimura isomorphism, we have 2M (τ, z) is the theta series defined by Formula (4.2). We define the Jacobi forms 2M (τ, z), g ≥ 0. Since f g ∈ [Γ g , k] is a Jacobi form of weight k and index 0, we get F g ∈ J ℓ,M (Γ g ) for all g ≥ 0. For [(τ, z]) ∈ Hyp g,h , F g (τ, z) = 0, g ≥ 0. By a simple calculation, we obtain Ψ g,M F g = F g−1 for all g ≥ 1.
Thus (F g ) g≥0 is a non-trivial stable Schottky-Jacobi form of weight ℓ and index M for the universal hyperelliptic locus Hyp ∞,h . This completes the proof of the above theorem(=Theorem 1.2).
We define the invariant µ(Q) of a quadratic form (Q, Λ) by We put 2M (τ, z), g ≥ 0. Then (F g ) g≥0 is a stable Schottky-Jacobi form of weight 1 2 (m + h) and index M for the universal hyperelliptic locus.

Final Remarks
In the final section, we make some remarks and present several open questions.
Remark 5.1. Let 2M be a positive even unimodular symmetric integral matrix of degree h. Assume that g + rank(M) > 2k + 1 and k ∈ Z + is even. We denote by C k,M be the vector space of stable Jacobi forms of weight k and index M. According to Theorem 2.2, the Siegel-Jacobi operator Ψ g,M : J k,M (Γ g ) −→ J k,M (Γ g−1 ) is an isomorphism, and hence we obtain From Formula (4.1), we see that Therefore from (SO1) in Section 2, we get the vanishing result: Thus k ≡ 0 (mod 4) if J k,M (Γ g ) = 0. According to Yang [28], any Jacobi form in J k,M (Γ g ) is singular. We note that any element in [Γ g , k − h 2 ] is a singular modular form (see [8,10]). Hence we conclude that C k,M is spanned by stable Jacobi forms of the form where P runs over the set of positive even unimodular quadratic forms of rank 2k − h.
Remark 5.2. Let ϕ g (τ ) be the Igusa modular form defined by the formula (3.2). We denote by [Γ g , k] 0 be the space of all Siegel cuspidal Hecke eigenforms on H g of weight k. It is known that [Γ 4 , 8] 0 = C · ϕ 4 (for a nice proof of this, we refer to [5]). Poor [20] showed that ϕ g (τ ) vanishes on the hyperelliptic locus Hyp g for all g ≥ 1, and the divisor of ϕ g (τ ) in A g is proper and irreducible for all g ≥ 4. And Ikeda [16,17] proved that if g ≡ k (mod 2), there exists a canonical lifting Considering the special cases of the Ikeda lift maps I 2,6 and I 6,6 , Breulman and Kuss [2] showed that I 2,6 (∆) = c ϕ 4 , c( = 0) ∈ C, and constructed a nonzero Siegel cusp form of degree 12 and weight 12 which is the image of ∆(τ ), where ∆(τ ) = (2πi) 12 q ∞ n=1 (1 − q n ) 24 , q := e 2πiτ , τ ∈ H 1 is a cusp form of weight 12.
Remark 5.3. We consider a half-integral semi-positive symmetric integral matrix M such that 2M is not even or which is not unimodular. The natural questions arise: and let B [4] (M) 1 be the subspace of all stable Schottky-Jacobi forms for the universal hyperelliptic locus contained in A [4] (M) 1 . Using Theorem 4.2 in [3], we can show that is a stable Schottky-Jacobi form for the universal hyperelliptic locus of weight 1 2 (m + h) and index M, that is, (Θ P − Θ Q ) Θ 2M ∈ B [4] (M) 1 . Here P and Q are two positive even unimodular quadratic forms of the same rank m (m ∈ Z + ), and Θ P , Θ Q are stable modular forms that are defined in Formula (3.1). The subspace B [4] (M) 1 of A [4] (M) 1 is spanned by all the stable Jacobi forms of type (5.1), where m runs over the set of all positive integers 8n (n ∈ Z + ).