Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On exceptional eigenvalues of the Laplacian for $ \Gamma _{0}(N)$

Author: Xian-Jin Li
Journal: Proc. Amer. Math. Soc. 136 (2008), 1945-1953
MSC (2000): Primary 11F37, 11F72
Published electronically: February 14, 2008
MathSciNet review: 2383500
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Abstract: An explicit Dirichlet series is obtained, which represents an analytic function of $ s$ in the half-plane $ \Re s>1/2$ except for having simple poles at points $ s_{j}$ that correspond to exceptional eigenvalues $ \lambda _{j}$ of the non-Euclidean Laplacian for Hecke congruence subgroups $ \Gamma _{0}(N)$ by the relation $ \lambda _{j}=s_{j}(1-s_{j})$ for $ j=1,2,\cdots , S$. Coefficients of the Dirichlet series involve all class numbers $ h_{d}$ of real quadratic number fields. But, only the terms with $ h_{d}\gg d^{1/2-\epsilon }$ for sufficiently large discriminants $ d$ contribute to the residues $ m_{j}/2$ of the Dirichlet series at the poles $ s_{j}$, where $ m_{j}$ is the multiplicity of the eigenvalue $ \lambda _{j}$ for $ j=1,2,\cdots , S$. This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on $ N$.

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Xian-Jin Li
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Keywords: Class numbers, Hecke operators, Maass wave forms, real quadratic fields
Received by editor(s): May 15, 2006
Received by editor(s) in revised form: March 5, 2007
Published electronically: February 14, 2008
Additional Notes: This research was supported by National Security Agency H98230-06-1-0061
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.