On Exceptional Eigenvalues of the Laplacian for Γ 0 ( N ) 3 2
نویسندگان
چکیده
Abstract. An explicit Dirichlet series is obtained, which represents an analytic function of s in the half-plane Rs > 1/2 except for having simple poles at points sj that correspond to exceptional eigenvalues λj of the non-Euclidean 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 ≫ d 1/2−ǫ 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 [3] the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on N .
منابع مشابه
A Dirichlet Series Related to Eigenvalues of the Laplacian for Congruence Subgroups
For any congruence subgroup Γ 0 (N), an explicit Dirichlet series is given which represents an analytic function of s in the half-plane Re s > 1/2 except for having simple poles at s = 1, 1/2 + p 1/4 − λ j , j = 1, 2, · · · , S, where λ j , j = 1, 2, · · · , S, are the exceptional eigenvalues of the non-Euclidean Laplacian for the congruence subgroup.
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