One-Level density for holomorphic cusp forms of arbitrary level

TL;DR

This paper extends the understanding of zero distributions of $L$-functions associated with holomorphic cusp forms to arbitrary levels, removing previous restrictions and connecting to random matrix theory predictions.

Contribution

It develops a trace formula for arbitrary level holomorphic cusp forms using a basis by Blomer and Milićević, generalizing prior results that required square-free levels.

Findings

Established one-level density matches random matrix theory predictions for arbitrary levels.

Removed the square-free level restriction in the analysis of $L$-function zeros.

Provided a new trace formula applicable to a broader class of automorphic forms.

Abstract

In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Mili\'cevi\'c, which is of use for other problems as well.