Fourier coefficients and cuspidality of modular forms: a new approach

TL;DR

This paper introduces a new induction-based method to distinguish cusp forms from Fourier coefficient growth in modular forms, achieving optimal weight ranges and proposing a related conjecture involving Fourier-Jacobi coefficients.

Contribution

It presents a novel, simple induction approach for identifying cusp forms, extending the range of weights and proposing a new conjecture based on Fourier-Jacobi coefficients.

Findings

Optimal weight ranges for cusp form identification

Partial validation of the Fourier-Jacobi coefficient conjecture

Method to recover cuspidality from Rankin-Selberg L-series poles

Abstract

We provide a simple and new induction based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and has wide adaptability. We propose a conjecture which asks the same converse question based on information on the Fourier-Jacobi coefficients, and answer it partially. We also discuss how to recover cuspidality from the poles of the allied Rankin-Selberg $L$-series.