On exceptional eigenvalues of the Laplacian for

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 in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean 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 .

**1.**J. Brian Conrey and Xian-Jin Li,*On the trace of Hecke operators for Maass forms for congruence subgroups*, Forum Math.**13**(2001), no. 4, 447–484. MR**1830242**, 10.1515/form.2001.017**2.**P. G. L. Dirichlet,*Lectures on number theory*, History of Mathematics, vol. 16, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1999. Supplements by R. Dedekind; Translated from the 1863 German original and with an introduction by John Stillwell. MR**1710911****3.**Henryk Iwaniec,*Spectral methods of automorphic forms*, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR**1942691****4.**Henry H. Kim,*Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂*, J. Amer. Math. Soc.**16**(2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR**1937203**, 10.1090/S0894-0347-02-00410-1**5.**Henry H. Kim and Freydoon Shahidi,*Functorial products for 𝐺𝐿₂×𝐺𝐿₃ and the symmetric cube for 𝐺𝐿₂*, Ann. of Math. (2)**155**(2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart. MR**1923967**, 10.2307/3062134**6.**Xian-Jin Li,*On the trace of Hecke operators for Maass forms*, Number theory (Ottawa, ON, 1996) CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 215–229. MR**1684604****7.**Xian-Jin Li,*On the trace of Hecke operators for Maass forms for congruence subgroups. II*, Forum Math.**17**(2005), no. 1, 1–30. MR**2110536**, 10.1515/form.2005.17.1.1**8.**W. Luo, Z. Rudnick, and P. Sarnak,*On Selberg’s eigenvalue conjecture*, Geom. Funct. Anal.**5**(1995), no. 2, 387–401. MR**1334872**, 10.1007/BF01895672**9.**Peter C. Sarnak,*Class numbers of indefinite binary quadratic forms. II*, J. Number Theory**21**(1985), no. 3, 333–346. MR**814010**, 10.1016/0022-314X(85)90060-5**10.**Atle Selberg,*On the estimation of Fourier coefficients of modular forms*, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1–15. MR**0182610****11.**C. L. Siegel,*Über die Classenzahl quadratischer Zahlkörper*, Acta Arith.**1**(1935), 83-86.

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Additional Information

**Xian-Jin Li**

Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602

Email:
xianjin@math.byu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-08-09151-X

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.