Low-Lying Zeros of $L$-functions of Ad\'elic Hilbert Modular Forms and their Convolutions

TL;DR

This paper investigates the distribution of low-lying zeros of $L$-functions associated with adelic Hilbert modular forms and their convolutions, providing evidence for the Katz-Sarnak density conjecture under GRH.

Contribution

It extends previous work by establishing new instances supporting the Katz-Sarnak conjecture for these $L$-functions under GRH and derives bounds on their average order and non-vanishing results.

Findings

Supports the Katz-Sarnak conjecture for these $L$-functions under GRH.

Provides an upper bound for the average order of $L$-functions at $s=1/2$.

Shows a positive proportion of non-vanishing of certain Rankin-Selberg $L$-functions.

Abstract

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances supporting the conjecture and extending the works of Iwaniec-Luo-Sarnak and many others. For applications, we obtain an upper bound for the average order of $L$-functions of Hilbert modular forms at $s=\frac{1}{2}$ as well as a positive proportion of non-vanishing of certain Rankin-Selberg $L$-functions.