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New bounds for fundamental Fourier coefficients of Siegel modular forms

Edgar Assing

Abstract

We prove new bounds for the Fourier coefficients of Jacobi forms using a method of Iwaniec. In view of the Fourier-Jacobi expansion of degree two Siegel modular forms, we can use these to obtain strong bounds on fundamental Fourier coefficients of Siegel modular forms.

New bounds for fundamental Fourier coefficients of Siegel modular forms

Abstract

We prove new bounds for the Fourier coefficients of Jacobi forms using a method of Iwaniec. In view of the Fourier-Jacobi expansion of degree two Siegel modular forms, we can use these to obtain strong bounds on fundamental Fourier coefficients of Siegel modular forms.

Paper Structure

This paper contains 10 sections, 13 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Reduction to Jacobi forms
  3. The method
  4. Kloosterman sums
  5. Jacobi forms
  6. The method of Iwaniec
  7. Setting up the scene
  8. The tails
  9. Estimation of $\mathcal{T}_{t,s}^{\pm}$
  10. The endgame

Key Result

Theorem 1.1

Let $k>2$ be even and let $F\in S^{(2)}_k(\textrm{Sp}_4(\mathbb{Z}))$. Then, for fundamental $T$ (i.e. $-4\det(T)$ is a fundamental discriminant), we have

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7: Kohnen 1993
  • Theorem 1.8
  • Remark 1.9
  • ...and 16 more