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

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 ${\mathrm{GL}}_2$ 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.

version 2.1, 19 pages, fixed some typos. To appear in IMRN

References in corpus (4)

• $n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions
• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
• On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of $L$-functions
• Maass waveforms and low-lying zeros

Cited by in corpus (6)

• Sato-Tate equidistribution for families of Hecke-Maass forms on SL(n,R)/SO(n)
• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
• On the moments of one-level densities in families of holomorphic cusp forms in the level aspect
• Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields
• Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values
• Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters