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

Jesse Jääsaari

TL;DR

The work addresses the fine-scale distribution of real zeros of half-integral weight Hecke cusp forms on $\Gamma_0(4)\backslash\mathbb{H}$ near the cusp, proving that many forms in the Kohnen plus subspace exhibit near-optimal counts of zeros along the lines $\Re(s)=-\tfrac{1}{2}$ and $\Re(s)=0$ in shrinking regions. The authors reduce zero detection to sign changes of Fourier coefficients via the Shimura correspondence and Waldspurger’s formula, and they control these sign changes through averaged first and second moments of quadratic twists $L(\tfrac{1}{2},f\otimes\chi_d)$, employing a half-integral weight Petersson formula, the Deshouillers–Iwaniec large sieve, and mollification techniques. The main results provide an almost-optimal lower bound for the number of real zeros for a positive proportion of forms and an unconditional lower bound for a large subset, with the latter derived from moment bounds and non-vanishing results. This advances the small-scale zero-distribution understanding for half-integral weight forms and extends Ghosh–Sarnak-type phenomena to the Kohnen plus setting, with potential further refinements via mollifier-based positive-density results.

Abstract

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $Γ_0(4)\backslash\mathbb H$ 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.

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

TL;DR

The work addresses the fine-scale distribution of real zeros of half-integral weight Hecke cusp forms on near the cusp, proving that many forms in the Kohnen plus subspace exhibit near-optimal counts of zeros along the lines and in shrinking regions. The authors reduce zero detection to sign changes of Fourier coefficients via the Shimura correspondence and Waldspurger’s formula, and they control these sign changes through averaged first and second moments of quadratic twists , employing a half-integral weight Petersson formula, the Deshouillers–Iwaniec large sieve, and mollification techniques. The main results provide an almost-optimal lower bound for the number of real zeros for a positive proportion of forms and an unconditional lower bound for a large subset, with the latter derived from moment bounds and non-vanishing results. This advances the small-scale zero-distribution understanding for half-integral weight forms and extends Ghosh–Sarnak-type phenomena to the Kohnen plus setting, with potential further refinements via mollifier-based positive-density results.

Abstract

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold 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 and as the weight tends to infinity. We show that, for of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large constant , 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 -functions.
Paper Structure (19 sections, 22 theorems, 169 equations)

This paper contains 19 sections, 22 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. The Strategy
  3. Possible extensions
  4. Organisation of the article
  5. Acknowledgements
  6. Notations
  7. Preliminaries
  8. Half-integral weight forms
  9. Tools from the theory of automorphic forms
  10. Other tools
  11. Reduction to the study of Fourier coefficients
  12. Proof of Lemma \ref{['second-moment-with-average']}
  13. Proof of Lemma \ref{['fourth-moment-with-average']}
  14. Proof of Theorem \ref{['main-theorem']}
  15. Proof of Proposition \ref{['Prop1']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $K$ be a large parameter, $\varepsilon>0$ be an arbitrarily small fixed number, and $j\in\{1,2\}$. Then for $\gg_\varepsilon K^2/(\log K)^{3/2+\varepsilon}$ of the forms $g\in\mathcal{S}_K$ we have for $\sqrt{K\log K}\leq Y\leq K^{1-\delta}$ with any small fixed constant $\delta>0$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • ...and 16 more