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arXiv · 1502.06885

Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level

Peter Humphries
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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 $Γ_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Stromberg, who proved this for $q = 9$ by different methods.

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Peter Humphries. 2017-12-09. Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level. https://doi.org/10.1093/imrn%2Frnx283

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