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 $j=1,2,\cdots , S$. 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 $j=1,2,\cdots , S$. 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 $N$.