Spectral Multiplicity for Maa{\ss} Newforms of Non-Squarefree Level

TL;DR

This paper demonstrates that for certain non-squarefree levels, a positive proportion of Maass newforms have Laplacian eigenvalues with multiplicities growing exponentially with the number of specific prime divisors, showing the spectrum's non-simplicity.

Contribution

It generalizes Stromberg's result by establishing eigenvalue multiplicities for a broad class of non-squarefree levels using spectral analysis.

Findings

Eigenvalues have multiplicity at least 2^{s(q)} for levels with s(q) odd prime divisors p where p^2 divides q

The cuspidal spectrum of the Laplacian on rac{rac{q}{ackslashmathbb{H}}} cannot be simple for any odd non-squarefree q

Positive proportion of eigenvalues exhibit multiplicity growth related to prime divisors

Abstract

We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maass newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $\Gamma_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Stromberg, who proved this for $q = 9$ by different methods.