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

TL;DR

This paper investigates the distribution of zeros of half-integral weight Hecke cusp forms near a cusp at infinity, showing that many zeros lie on specific vertical geodesics as the weight increases, with results supported by moments of quadratic twists of L-functions.

Contribution

It provides the first quantitative results on the distribution of zeros of half-integral weight Hecke cusp forms near a cusp, confirming conjectured behavior for a large subset of forms.

Findings

Almost all zeros near the cusp lie on two vertical geodesics for a large subset of forms.

Established asymptotic evaluation of moments of quadratic twists of modular L-functions.

Proved a positive proportion of forms have a significant number of real zeros.

Abstract

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $\Gamma_0(4)\backslash\mathbb H$ near a cusp at infinity. In analogue of the Ghosh-Sarnak conjecture for classical holomorphic Hecke cusp forms, one expects that almost all of the zeroes sufficiently close to this cusp lie on two vertical geodesics $\Re(s)=-1/2$ and $\Re(s)=0$ as the weight tends to infinity. We show that, for $\gg_\varepsilon K^2/(\log K)^{3/2+\varepsilon}$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large constant $K$, the number of such "real" zeroes grows almost at the expected rate. We also obtain a weaker lower bound for the number of real zeroes that holds for a positive proportion of forms. One of the key ingredients is the asymptotic evaluation of averaged first and second moments of quadratic twists of modular $L$-functions.