On exceptional eigenvalues of the Laplacian for $\Gamma_0(N)$

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 $h_d\gg d^{1/2-\epsilon}$ 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 $N$.