Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weighted low-lying zeros of L-functions attached to Siegel modular forms

Shifan Zhao

TL;DR

This work studies weighted low-lying zeros of spinor and standard $L$-functions attached to degree $2$ Siegel modular forms within the Katz–Sarnak density framework. By combining an explicit formula with a Siegel Petersson formula (including an averaged version over weight), the authors prove that the weighted zeros exhibit symplectic symmetry for test functions with Fourier support up to $(-1,1)$ for spinor $L$-functions and, after averaging, up to $(-5/18,5/18)$ for standard $L$-functions. The spinor case yields a direct non-vanishing corollary: $ ext{liminf}_{k o ty} rac{ ext{sum of }oldsymbol{ ext{ω}_F}}{ ext{weighted count}} ext{ of }L(1/2,F; ext{spin}) eq0 \

Abstract

In this paper, we study weighted low-lying zeros of spinor and standard $L$-functions attached to degree 2 Siegel modular forms. We show the symmetry type of weighted low-lying zeros of spinor $L$-functions is symplectic, for test functions whose Fourier transform have support in $(-1,1)$, extending the previous range $(-\frac{4}{15},\frac{4}{15})$ by E. Kowalski, A. Saha and J. Tsimerman . We then show the symmetry type of weighted low-lying zeros of standard $L$-functions is also symplectic. We further extend the range of support by performing an average over weight. As an application, we discuss non-vanishing of central values of those $L$-functions.

Weighted low-lying zeros of L-functions attached to Siegel modular forms

TL;DR

This work studies weighted low-lying zeros of spinor and standard -functions attached to degree Siegel modular forms within the Katz–Sarnak density framework. By combining an explicit formula with a Siegel Petersson formula (including an averaged version over weight), the authors prove that the weighted zeros exhibit symplectic symmetry for test functions with Fourier support up to for spinor -functions and, after averaging, up to for standard -functions. The spinor case yields a direct non-vanishing corollary: $ ext{liminf}_{k o ty} rac{ ext{sum of }oldsymbol{ ext{ω}_F}}{ ext{weighted count}} ext{ of }L(1/2,F; ext{spin}) eq0 \

Abstract

In this paper, we study weighted low-lying zeros of spinor and standard -functions attached to degree 2 Siegel modular forms. We show the symmetry type of weighted low-lying zeros of spinor -functions is symplectic, for test functions whose Fourier transform have support in , extending the previous range by E. Kowalski, A. Saha and J. Tsimerman . We then show the symmetry type of weighted low-lying zeros of standard -functions is also symplectic. We further extend the range of support by performing an average over weight. As an application, we discuss non-vanishing of central values of those -functions.
Paper Structure (13 sections, 7 theorems, 134 equations)

This paper contains 13 sections, 7 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Spinor and Standard L-functions
  3. The Spinor L-function
  4. The Standard L-function
  5. Combinatorial Relations
  6. The Petersson Formula
  7. Proof of Main Theorems
  8. Proof of Theorem 1.1
  9. Proof of Theorem 1.2
  10. Proof of Theorem 1.3
  11. Application to Non-vanishing of Central Values
  12. Proof of Corollary 1.1
  13. Further Discussion

Key Result

Theorem 1.1

Let $\Phi$ be an even Schwartz function whose Fourier transform has support in $(-1,1)$. For $F \in H_k(\Gamma_2)$, define $D(F;\Phi;\text{spin})$ as in density spinor L-function with $c_{F;\text{spin}} = k^2$ and $\omega_F$ as in harmonic weight. Assume GRH for $L(s,F;\text{spin})$. Then we have

Theorems & Definitions (17)

  • Conjecture 1.1: Density Conjecture
  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.3
  • Lemma 2.1
  • Remark 2.1
  • ...and 7 more