Bounds for Siegel Modular Forms of genus 2 modulo ppp (original) (raw)
manuscripta mathematica, 2021
A. We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms mod ℓ, partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish mod ℓ. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients mod ℓ of a Siegel modular form with integral algebraic Fourier coefficients provided ℓ is large enough. We also make some efforts to make this "largeness" of ℓ effective.
A note on Fourier-Jacobi coefficients of Siegel modular forms
Archiv der Mathematik, 2013
Let F be a Siegel cusp form of weight k and genus n > 1 with Fourier-Jacobi coefficients f m. In this article, we estimate the growth of the Petersson norms of f m , where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen in [5].
On Fundamental Fourier Coefficients of Siegel Modular Forms
Journal of the Institute of Mathematics of Jussieu, 2021
We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree 333 and level 111 .
Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2
arXiv (Cornell University), 2023
Let F be an L 2-normalized Siegel cusp form for Sp 4 (Z) of weight k that is a Hecke eigenform and not a Saito-Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound |a(F, S)| ≪ǫ k 1/4+ǫ (4π) k Γ(k) c(S) − 1 2 det(S) k−1 2 +ǫ where c(S) denotes the gcd of the entries of S, and that its global sup-norm satisfies the bound (det Y) k 2 F ∞ ≪ǫ k 5 4 +ǫ. The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.
On congruences for the coefficients of modular forms and some applications
1997
We start with a brief overview of the necessary theory: Given any cusp form f=∑ n≥ 1 an (f) qn of weight k, we denote by L (f, s) the L-function of f. For Re (s)> k/2+ 1, the value of L (f, s) is given by L (f, s)=∑ n≥ 1 an (f) ns and, one can show that L (f, s) has analytic continuation to the entire complex plane. The value of L (f, s) at s= k/2 will be of particular interest to us, and we will refer to this value as the central critical value of L (f, s).
Fourier coefficients of half-integral weight modular forms modulo ell
1996
For each prime ℓ, let |·|_ℓ be an extension to of the usual ℓ-adic absolute value on . Suppose g(z) = ∑_n=0^∞ c(n)q^n ∈ M_k+(N) is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes ℓ there are infinitely many square-free integers m for which |c(m)|_ℓ = 1. Consequently we obtain indivisibility results for "algebraic parts" of central critical values of modular L-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for L-function values. For example if Δ(z) is Ramanujan's cusp form and g(z)=∑_n=1^∞c(n)q^n is the cusp form for which L(Δ_D,6)=()πD^6...
Prime number theorems for the coefficients of modular forms
Bulletin of The American Mathematical Society, 1972
Communicated by P. T. Bateman, February 25, 1972 2. Statement of results. As in the classical situation of the Riemann zeta function, the following results lead to an improvement of (3). LEMMA 1 (GENERALIZED VON MANGOLDT FORMULA). Let N^ (7) denote the number of zeros p = j8 + iy of(p(s) with 11/2 ^ p ^ 13/2 and 0 ^ y ^ T.
Acta Arithmetica, 1999
Dedicated to the memory of my greatest teachers, my parents, Thereza de Azevedo Pribitkin and Edmund Pribitkin 1. Historical introduction. In 1989 Knopp [6] found explicit formulas for the Fourier coefficients of an arbitrary cusp form and more generally, but conditionally, of a holomorphic modular form (with a possible pole at i∞) on the full modular group, Γ (1), of weight k, 4/3 < k < 2, and multiplier system v. He assumed that there are no nontrivial cusp forms on Γ (1) of complementary weight 2−k and conjugate multiplier system v. In our initial paper we remove this assumption and capture the Fourier coefficients of an arbitrary "Niebur modular integral" on Γ (1) of weight k, 1 < k < 2. En route we also obtain expressions for the Fourier coefficients of an arbitrary cusp form on Γ (1) of weight k, 0 < k < 1. In particular we present formulas for the Fourier coefficients of η r (τ), 0 < r < 2, where η(τ) is the Dedekind eta-function. An actual formula for the Fourier coefficients of an arbitrary modular form, even in the case of the full modular group, is not always available. For forms of weight greater than two the problem was solved by Petersson [11], who introduced the (parabolic) Poincaré series. Additionally, by considering a nonanalytic version of this series, he derived the coefficients of certain forms of weight two [12]. By integrating one of these forms, Petersson [12, p. 202] was the first to find the coefficients of the absolute modular invariant J(τ). For forms of negative weight Rademacher and Zuckerman [18] discovered expressions for the coefficients by relying on the circle method. Furthermore, Rademacher [15] employed a sharpened version of this method to rediscover Petersson's formula for J(τ). We remark that both approaches