On fundamental Fourier coefficients of Siegel modular forms

TL;DR

This paper proves that non-zero vector-valued Siegel modular forms have infinitely many non-zero Fourier coefficients with fundamental discriminants, and applies this to establish an unconditional functional equation for certain spinor L-functions.

Contribution

It establishes the existence of infinitely many fundamental discriminant Fourier coefficients for any non-zero Siegel modular form and applies this to prove a key functional equation unconditionally.

Findings

Existence of infinitely many non-zero Fourier coefficients with fundamental discriminants.

Application to an unconditional proof of the functional equation for degree 3 spinor L-functions.

Use of induction in the setting of vector-valued modular forms.

Abstract

We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero 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. In an Appendix, as an application of a variant of our result and building upon 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 $3$.