Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


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
Published electronically: April 3, 2014
MathSciNet review: 3246823
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11F72, 58C40, 34L16

Retrieve articles in all journals with MSC (2010): 11F72, 58C40, 34L16

Additional Information

Jay Jorgenson
Affiliation: Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031

Lejla Smajlović
Affiliation: Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina

Holger Then
Affiliation: Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom

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