On the distribution of eigenvalues of Maass forms on certain moonshine groups

TL;DR

This paper investigates the eigenvalue distribution of Maass forms on moonshine groups, deriving Weyl's law, comparing groups with similar signatures, and numerically verifying eigenvalue distribution conjectures.

Contribution

It provides the first proof of Weyl's law for these groups, compares eigenvalue counts for different groups, and employs advanced algorithms to empirically analyze eigenvalue distributions.

Findings

Asymptotic difference in cusp forms between groups with same signature

Unconditional proof of Weyl's law for moonshine groups

Numerical verification of eigenvalue distribution conjectures

Abstract

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $\Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $\Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $\Gamma_0(5)^+$ than for $\Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $\Gamma_0(5)^+$ and the first $12474$ eigenvalues of $\Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.