Low lying zeros of families of L-functions

TL;DR

This paper studies the distribution of zeros near the central point for families of GL_2 automorphic L-functions, providing sharper nonvanishing results under the assumption of the Generalized Riemann Hypothesis.

Contribution

It offers new insights into the zeros distribution near s=1/2 for these L-functions, conditional on GRH, improving nonvanishing estimates compared to previous work.

Findings

Conditional results on zeros distribution near s=1/2

Sharper nonvanishing estimates under GRH

Analysis specific to families of GL_2 automorphic L-functions

Abstract

In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.