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

  title={On exceptional eigenvalues of the Laplacian for \$\Gamma\_0(N)\$},
  author={Xian-jin Li},
  journal={arXiv: Number Theory},
  • Xian-jin Li
  • Published 3 October 2006
  • Mathematics
  • arXiv: Number Theory
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… 
The Selberg trace formula as a Dirichlet series
We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in


Functorial products for $\mathrm{GL}_2 \times \mathrm{GL}_3$ and the symmetric cube for $\mathrm{GL}_2$
In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on GL2 × GL3 as automorphic forms on GL6, from which we obtain our second case, the
On Selberg's eigenvalue conjecture
Let Γ ⊂ SL 2(Z) be a congruence subgroup, and λ0 = 0 3/16. Iwaniec ([I]) showed that for almost all Hecke congruence groups Γ0(p) with a certain multiplier χ p , one has λ1(Γ0(p), χ p ) ≥ 44/225 =
On the Trace of Hecke Operators for Maass Forms for Congruence Subgroups II
Abstract Let E λ be the Hilbert space spanned by the eigenfunctions of the non-Euclidean Laplacian associated with a positive discrete eigenvalue λ. In this paper, the trace of the Hecke operators
Lectures on Number Theory
On the divisibility of numbers On the congruence of numbers On quadratic residues On quadratic forms Determination of the class number of binary quadratic forms Some theorems from Gauss's theory of
Spectral methods of automorphic forms
Introduction Harmonic analysis on the Euclidean plane Harmonic analysis on the hyperbolic plane Fuchsian groups Automorphic forms The spectral theorem. Discrete part The automorphic Green function