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


Authors: Jay Jorgenson, Lejla Smajlović and Holger Then
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
Published electronically: April 3, 2014
MathSciNet review: 3246823
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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.


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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
Email: jjorgenson@mindspring.com

Lejla Smajlović
Affiliation: Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina
Email: lejlas@pmf.unsa.ba

Holger Then
Affiliation: Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom
Email: holger.then@bristol.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2014-02823-8
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.
Article copyright: © Copyright 2014 American Mathematical Society