On exceptional eigenvalues of the Laplacian for $Γ_0(N)$
| dc.creator | Li, Xian-Jin | |
| dc.date | 2006-10-03 | |
| dc.date.accessioned | 2026-07-07T07:28:40Z | |
| dc.date.available | 2026-07-07T07:28:40Z | |
| dc.description | 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 $λ_j$ of the non-Euclidean Laplacian for Hecke congruence subgroups $Γ_0(N)$ by the relation $λ_j=s_j(1-s_j)$ 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 $h_d\gg d^{1/2-ε}$ 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 $λ_j$ 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 $N$. | |
| dc.identifier | https://arxiv.org/abs/math/0610120 | |
| dc.identifier | http://arxiv.org/abs/math/0610120 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/117816 | |
| dc.subject | Number Theory | |
| dc.subject | 11F37, 11F72 | |
| dc.title | On exceptional eigenvalues of the Laplacian for $Γ_0(N)$ | |
| dc.type | text |