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

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.

arXiv comment21 pages

References in corpus (6)

• Low-lying zeros of families of elliptic curves
• Local spectral equidistribution for Siegel modular forms and applications
• The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions
• Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture
• One level density of low-lying zeros of families of $L$-functions
• A weighted one-level density of families of $L$-functions