Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields

TL;DR

This paper investigates the distribution of low-lying zeros of $L$-functions associated with Hecke--Maass forms over imaginary quadratic fields, showing they match orthogonal ensembles in certain limits.

Contribution

It establishes the correspondence between the low-lying zeros of these $L$-functions and orthogonal random matrix models for the first and second level densities.

Findings

$1$-level density matches orthogonal ensemble for square-free level as norm tends to infinity.

Similar results for full level forms as Laplace eigenvalues grow.

Supports the universality of random matrix theory in number theory contexts.

Abstract

We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that the $1$-level density for Hecke--Maass forms over $F$ of square-free level $\mathfrak{q}$, as $\mathrm{N}(\mathfrak{q})$ tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke--Maass forms over $F$ of full level, we prove similar statements for the $1$- and $2$-level densities, as the Laplace eigenvalues tends to infinity.