Low-lying zeroes of Maass form $L$-functions

TL;DR

This paper proves that the low-lying zeros of level 1 Maass form $L$-functions follow the orthogonal distribution, extending the understanding of zero distributions in automorphic $L$-functions beyond cusp forms.

Contribution

It establishes the one-level density for low-lying zeros of Maass forms with test functions supported in a specific range, confirming orthogonal symmetry unconditionally.

Findings

Low-lying zeros of Maass form $L$-functions follow orthogonal distribution.

Supports larger test function Fourier transforms than previous results.

Unconditional proof of orthogonal symmetry for Maass forms.

Abstract

The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups $U(N)$. This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight $k$ and level $N$. They showed in the limit as $kN \to\infty$ that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in $(-2,2)$. Exceeding $(-1,1)$ is important as the three orthogonal groups are indistinguishable for support up to $(-1,1)$ but are distinguishable for any larger support. We study the other family of ${\rm GL}_2$ automorphic forms over $\mathbb{Q}$: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order $M$ at the origin. For test functions with Fourier transform supported inside $\left(-2 + \frac{2}{2M+1}, 2 - \frac{2}{2M+1}\right)$, we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.