On exceptional eigenvalues of the Laplacian for
Author:
XianJin Li
Journal:
Proc. Amer. Math. Soc. 136 (2008), 19451953
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 in the halfplane except for having simple poles at points that correspond to exceptional eigenvalues of the nonEuclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . 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 .
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Additional Information
XianJin Li
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email:
xianjin@math.byu.edu
DOI:
http://dx.doi.org/10.1090/S000299390809151X
PII:
S 00029939(08)09151X
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 H982300610061
Communicated by:
WenChing Winnie Li
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
