On Petersson products of not necessarily cuspidal modular forms (original) (raw)
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On Dirichlet Series and Petersson Products for Siegel Modular Forms
Annales de l’institut Fourier, 2008
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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].
Petersson norms of not necessarily cuspidal Jacobi modular forms and applications
Advances in Mathematics, 2018
We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on H 2 and those on H 1 × C (here H n denotes the Siegel upper half space of degree n). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms; and as a byproduct we also show that any Siegel modular form of degree 2 is determined by its 'fundamental' Fourier coefficients.
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).
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
On generalized modular forms and their applications
Nagoya Math. J, 2008
Abstract. We study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain ...