Sturm bounds for Siegel modular forms

Abstract

We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g = 1, 2, which do not extend to the case of general g. MSC 2010: Primary 11F46; Secondary 11F33Let p be a prime. A celebrated theorem of Sturm [13] implies that an elliptic modular form with p-integral rational Fourier series coefficients is determined by its "first few" Fourier series coefficients modulo p. Sturm's theorem is an important tool in the theory of modular forms (for example, see [7,12] for some of its applications). Poor and Yuen [9] (and later [2] for p ≥ 5) proved a Sturm theorem for Siegel modular forms of degree 2. Their work has been applied in different contexts, and for example, it allowed [3,4] to confirm Ramanujan-type congruences for specific Siegel modular forms of degree 2. In [10], we gave a characterization of U(p) congruences of Siegel modular forms of arbitrary degree, but (lacking a Sturm theorem) we could only discuss one explicit example that occurred as a Duke-Imamoglu-Ikeda lift. If a Siegel modular form does not arise as a lift, then one needs a Sturm theorem to justify its U(p) congruences.In this paper, we provide such a Sturm theorem for Siegel modular forms of degree g ≥ 2. Our proof is totally different from the proofs of the cases g = 1, 2 in [2,9,13], which do not have visible extensions to the case g > 2. More precisely, we perform an induction on the degree g. As in [1], we employ Fourier-Jacobi expansions of Siegel modular forms, and we study vanishing orders of Jacobi forms. However, in contrast to [1] we consider restrictions of Jacobi forms to torsion points (instead of their theta decompositions), which allow us to relate mod p diagonal vanishing orders (defined in the first Section) of Jacobi forms and Siegel modular forms. We deduce the following theorem. Theorem I. Let F be a Siegel modular form of degree g ≥ 2, weight k, and with p-integral rational Fourier series coefficients c(T).Suppose that c(T) ≡ 0 (mod p) for all T = (t ij ) with t ii ≤ 4 3 g k 16 .Then c(T) ≡ 0 (mod p) for all T.

The isospectral problem for flat tori from three perspectives

“…In terms of quadratic forms, there is a connection to Siegel modular forms; see [45,Chapter 1]. For a certificate similar to Corollary 3.7 in terms of k-spectra, see [46]. We hope that readers investigating these and related problems will keep all three perspectives in mind and thereby reap the benefits of techniques from analysis, number theory, and geometry.…”

On mod p singular modular forms

“…In general, "Sturm bounds" give an explicit finite set of T ∈ Λ n such that a modular form of degree n with Fourier coefficients in Z (p) must be congruent mod p to zero if the Fourier coefficients for all T in that finite set are divisible by p, see e.g. [20,25]. Such an explicit finite set would usually contain quadratic forms of all ranks.…”

Notes on theta series for Niemeier lattices

“…In this section, we introduce a result of Richter and Raum [9] concerning the socalled Sturm bound. From this result, we can prove the p-divisibility or integrality of a modular form.…”

“…Ozeki [5] calculated the value a(ϑ (n) ω ; T ) for various degrees n [5, Table 4,5,6]. We list the congruence relations expected from his tables.…”