# Nonvanishing of central $L$-values of Maass forms

## TL;DR

This paper proves that a positive proportion of Maass forms' central L-values do not vanish as the spectral parameter grows large, using moments and mollification techniques, with applications to Fourier coefficients of half-integer weight forms.

## Contribution

It establishes a quantitatively optimal nonvanishing result for central L-values of Maass forms and applies it to Fourier coefficients of half-integer weight Maass forms.

## Key findings

- Positive proportion of nonvanishing central L-values in large spectral aspect
- Optimal nonvanishing proportion consistent with Weyl's law
- Nonvanishing of Fourier coefficients in Kohnen plus space

## Abstract

With the method of moments and the mollification method, we study the central $L$-values of GL(2) Maass forms of weight $0$ and level $1$ and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. As an application of this result and a formula of Katok--Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight $\frac{1}{2}$ and level $4$ in the Kohnen plus space.