On the trace of Hecke operators for Maass forms for congruence subgroups. (Q2716427)

Let \(N>1\) be a positive integer and let \(T_n\), \(n=1,2,\ldots\), \((n,N)=1\) be the Hecke operators, which act in the space of automorphic functions with respect to \(\Gamma_0(N)\) by NEWLINE\[NEWLINE(T_nf)(z)=\frac{1}{\sqrt{n}}\sum_{ad=n,0\leq b<d}f\left(\frac{az+b}{d}\right).NEWLINE\]NEWLINE Let \(E_\lambda\) be a Hilbert space of functions spanned by the eigenfunctions of the Laplace-Beltrami operator with a positive eigenvalue \(\lambda\). The authors compute the trace \(\operatorname{tr}T_n\) on the space \(E_\lambda\) for squarefree \(N\). Let \(h_d\) be the class number of indefinite rational quadratic forms with discriminant \(d\), let NEWLINE\[NEWLINE\varepsilon_d=\frac{v_0+u_0\sqrt{d}}{2}NEWLINE\]NEWLINE where the pair \((v_0,u_0)\) is the fundamental solution of Pell's equation \(v^2-du^2=4\) and let \(\Omega\) be the set of all the positive integers \(d\) such that \(d\equiv 0\) or \(1\pmod 4\) and such that \(d\) is not a square of an integer.NEWLINENEWLINETheorem. Let \(N>1\) be a squarefree integer, and let \(n\geq 1\) with \((n,N)=1\). Put NEWLINE\[NEWLINEL_n(s)=\sum_{m| N}\sum_{k| N}k^{1-2s}\frac{\mu((m,k))}{(m,k)} \sum_{d\in\Omega}\sum_u\left(\frac{d}{m}\right) \frac{h_d\log\varepsilon_d}{(du^2)^s}NEWLINE\]NEWLINE for \(\Re s>1\), where the summation on \(u\) is taken over all the positive integers \(u\) such that \(\sqrt{4n+dk^2u^2}\in\mathbb Z\). Then \(L_n(s)\) is analytic for \(\Re s>1\) and can be extended by analytic continuation to the half plane \(\Re s>0\) except for a possible pole at \(s=\frac{1}{2}\) and for possible simple poles at \(s=1,\frac{1}{2}\pm ik_j, j=1,2,\ldots.\) For any eigenvalue \(\lambda>0\) of the Laplace-Beltrami operator for \(\Gamma_0(N)\), we have NEWLINE\[NEWLINE\operatorname{tr}T_n=2n^{ik}\Re s_{s=\frac{1}{2}+ik}L_n(s),NEWLINE\]NEWLINE where \(k=\sqrt{\lambda-\frac{1}{4}}.\)