Low-lying zeros in families of Maass form L-functions: an extended density theorem

TL;DR

This paper extends the known support for the one-level density of low-lying zeros in Maass form L-functions, using advanced analytic techniques, and explores implications under the Grand Density Conjecture and for holomorphic forms.

Contribution

It unconditionally extends the support to (-15/8, 15/8) and conditionally to (-2, 2), improving previous results and applying methods to holomorphic form families.

Findings

Unconditional support extended to (-15/8, 15/8).

Conditional support extended to (-2, 2) under the Grand Density Conjecture.

Improved support results also apply to holomorphic form L-functions.

Abstract

We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.