Determination of modular forms by fundamental Fourier coefficients

TL;DR

This paper investigates when specific subsets of Fourier coefficients uniquely determine certain modular forms, focusing on half-integral weight and genus 2 Siegel modular forms, with connections to automorphic L-functions.

Contribution

It establishes conditions under which Fourier coefficients suffice to identify modular forms, linking classical and Siegel cases and relating to automorphic L-functions.

Findings

Fourier coefficients can uniquely determine certain modular forms under specific conditions.

The study reveals a close relationship between half-integral weight forms and genus 2 Siegel modular forms.

Connections to automorphic L-functions and Bessel models are explored.

Abstract

This article is a research exposition based on the author's talk at the International Colloquium on Automorphic Representations and L-Functions, 2012, held at TIFR, Mumbai. We consider some special cases of the following question: when is a natural subset of the Fourier coefficients sufficient to uniquely determine a "modular form"? Two kinds of modular forms are considered in this article: a) classical modular forms of half-integral weight, and b) Siegel modular forms of genus 2 and integral weight. These two apparently different scenarios turn out to be closely related. Our results were motivated by, and have several interesting connections to, automorphic L-functions and Bessel models.