On the Real Zeroes of Half-integral Weight Hecke Cusp Forms, II

TL;DR

This paper proves that a large subset of half-integral weight Hecke cusp forms have the expected number of real zeros, using sharp bounds on moments of quadratic twists of modular L-functions.

Contribution

It establishes the growth rate of real zeros for half-integral weight Hecke cusp forms and introduces sharp bounds for moments of quadratic twists as a key technical tool.

Findings

Number of real zeros grows at the expected rate for most forms

Sharp bounds obtained for mollified moments of quadratic twists

Results apply to forms with weight up to a large parameter K

Abstract

We show that for $\gg K^2$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter $K$, the number of "real" zeroes grows at the expected rate. A key technical step in the proof is to obtain sharp bounds for the mollified first and second moments of quadratic twists of modular $L$-functions.