On the distribution of eigenvalues of Maass forms on certain moonshine groups
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- by Jay Jorgenson, Lejla Smajlović and Holger Then PDF
- Math. Comp. 83 (2014), 3039-3070 Request permission
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 $N$ is a positive, 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 [H. Then, Computing large sets of consecutive Maass forms, in preparation], and numerically determine the first $3557$ eigenvalues 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.References
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Additional Information
- Jay Jorgenson
- Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031
- MR Author ID: 292611
- Email: jjorgenson@mindspring.com
- Lejla Smajlović
- Affiliation: Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71 000 Sarajevo, Bosnia and Herzegovina
- ORCID: 0000-0002-2709-5535
- Email: lejlas@pmf.unsa.ba
- Holger Then
- Affiliation: Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom
- MR Author ID: 742378
- ORCID: 0000-0002-0368-639X
- Email: holger.then@bristol.ac.uk
- Received by editor(s): September 24, 2012
- Received by editor(s) in revised form: March 18, 2013
- Published electronically: April 3, 2014
- Additional Notes: The first author acknowledges grant support from NSF and PSC-CUNY grants
The second author acknowledges support from EPSRC grant EP/H005188/1. - © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 83 (2014), 3039-3070
- MSC (2010): Primary 11F72, 58C40; Secondary 34L16
- DOI: https://doi.org/10.1090/S0025-5718-2014-02823-8
- MathSciNet review: 3246823