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

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 Weyls law for the distribution of eigenvalues of Maass forms, from which we prove the classical Weyls 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 Weyls laws. In addition, we employ Hejhals 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.

On the appearance of Eisenstein series through degeneration

Let $Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $mathbb H$, and let $M = Gamma backslash mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $gamma$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $gamma in Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.