On exceptional eigenvalues of the Laplacian for Γ₀(N)

2006/10/03 by Xian-Jin Li, Li, Xian-Jin
Mathematics · #11F37 #11F72 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F37 #msc:11F72

paper · pdf · doi:10.48550/arxiv.math/0610120

arxiv created 2006/10/03 · arxiv updated 2009/12/01

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 λ_j of the non-Euclidean Laplacian for Hecke congruence subgroups Γ₀(N) by the relation λ_j=s_j(1-s_j) for j=1,2,..., S. Coefficients of the Dirichlet series involve all class numbers h_d of real quadratic number fields. But, only the terms with h_d≫ d^1/2-ε 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 λ_j for j=1,2,..., 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 [3] the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on N.

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