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Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level
International Mathematics Research Notices ( IF 1.291 ) Pub Date : 2017-12-08 , DOI: 10.1093/imrn/rnx283
Peter Humphries

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 Strömberg who proved this for |$q = 9$| by different methods.
更新日期:2020-04-17

 

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