On Exceptional Eigenvalues of the Laplacian for Γ0(n)
نویسندگان
چکیده
An explicit Dirichlet series is obtained, which represents an analytic function of s in the half-plane s > 1/2 except for having simple poles at points sj that correspond to exceptional eigenvalues λj of the nonEuclidean Laplacian for Hecke congruence subgroups Γ0(N) by the relation λj = sj(1− sj) for j = 1, 2, · · · , S. Coefficients of the Dirichlet series involve all class numbers hd of real quadratic number fields. But, only the terms with hd d1/2− for sufficiently large discriminants d contribute to the residues mj/2 of the Dirichlet series at the poles sj , where mj is the multiplicity of the eigenvalue λj 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 the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on N .
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On Exceptional Eigenvalues of the Laplacian for Γ 0 ( N ) 3 2
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