Maass waveforms and low-lying zeros

TL;DR

This paper extends the understanding of the distribution of zeros of Maass form L-functions near the central point, showing they follow orthogonal symmetry patterns similar to holomorphic cusp forms, for certain test functions and levels.

Contribution

It proves that the zeros of Maass form L-functions exhibit orthogonal symmetry behavior, expanding previous results from holomorphic cusp forms to Maass forms using the Petersson formula.

Findings

Zeros of Maass form L-functions match orthogonal group eigenvalue distributions.

Results hold for test functions with support in (-3/2, 3/2) as level N tends to infinity.

The symmetry type is confirmed via 2-level density calculations.

Abstract

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of $L$-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions $\phi$. We prove similar results for families of cuspidal Maass forms, the other natural family of ${\rm GL}_2/\mathbb{Q}$ $L$-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for $\Supp(\widehat{\phi})\subseteq (-3/2, 3/2)$ when the level $N$ tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in $(-1,1)$, though we still uniquely specify the symmetry type by computing the 2-level density.