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On exceptional eigenvalues of the Laplacian for $Gamma_0(N)$ | | Xian-Jin Li
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3 Oct 2006 | | Subject: | Number Theory | | 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 $lambda_j$ of the non-Euclidean Laplacian for Hecke congruence subgroups $Gamma_0(N)$ by the relation $lambda_j=s_j(1-s_j)$ 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 $h_dgg d^{1/2-epsilon}$ 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 $lambda_j$ 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 $N$. | | Source: | arXiv, math/0610120 | | Services: | Forum | Review | PDF | Favorites | |
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