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

TL;DR

This work establishes that real zeros of half-integral weight Hecke cusp forms in the Kohnen plus subspace occur with the expected density for a positive proportion of forms as the weight grows. The authors design a mollifier $M_g(d)$ to stabilize the Fourier coefficients and reduce the problem to sign changes of $c_g(|d|)$, then prove sharp mollified moments for quadratic twists $L(\tfrac12,f\otimes\chi_d)$. A novel random-model framework is developed to predict and control the mollified fourth moment, enabling precise diagonal/off-diagonal analysis and the extraction of the main term. Consequently, for $\gg K^2$ forms with $k\sim K$, the number of real zeros on the relevant geodesics grows like $K/Y$ in the regime $\sqrt{K\log K}\le Y\le K^{1-\vartheta}$, with implications for the fine-scale zero distribution of half-integral weight cusp forms.

Abstract

We show that for $\gg K^2$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspace 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.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Proposition 4.1
• Proposition 4.2
• Lemma 5.1
• Remark 5.2
• Lemma 5.3
• Lemma 5.4