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

Shifan Zhao

Vol. 335 (2025), No. 1, 183–210

DOI: 10.2140/pjm.2025.335.183

Abstract

We study weighted low-lying zeros of spinor and standard $L$-functions attached to degree 2 Siegel modular forms. We show that 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 $\left(-\frac{4}{15},\frac{4}{15}\right)$. We then show that 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 nonvanishing of central values of those $L$-functions.

Keywords

low-lying zeros, spinor $L$-functions, standard $L$-functions, Kitaoka's formula, nonvanishing

Mathematical Subject Classification

Primary: 11F46, 11F66, 11F72

Milestones

Received: 13 October 2024

Revised: 6 March 2025

Accepted: 9 March 2025

Published: 8 April 2025