On the non-vanishing of modular $L$-values and fourier coefficients of cusp forms

TL;DR

This paper proves non-vanishing results for modular L-values with quadratic twists in short intervals and uses these to improve understanding of Fourier coefficients of half-integral weight eigenforms.

Contribution

It introduces new non-vanishing results for modular L-values in short intervals and enhances previous bounds on Fourier coefficients of half-integral weight eigenforms.

Findings

Non-vanishing of modular L-values with quadratic twists in short intervals

Improved bounds on Fourier coefficients of half-integral weight eigenforms

Application of Waldspurger's theorem to strengthen non-vanishing results

Abstract

We prove a non-vanishing result of modular L-values with quadratic twists, where the quadratic discriminants are in a short interval. Using this fact and Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev density theorem in short intervals and some questions of Serre, Journal of number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier coefficients of half-integral weight eigenform.