On Cartier-Voros type Selberg trace formula for congruence subgroups of \(\text{PSL} (2,\mathbb{R})\)
Let \(\Gamma\) be a discrete subgroup of \(\text{PSL}(2,\mathbb{R})\) acting on the upper half plane \(H\) by fractional linear transformation. In particular, the author considers congruence subgroups of the modular group \(\text{PSL}(2,\mathbb{Z})\) so that the fundamental domain \(\Gamma\setminus H\) is non-compact but has finite volume. The object is to develop a trace formula to deal with the theta type function \(\Theta_\Gamma(t)= \sum^\infty_{n=0} e^{-t\rho_n}\), where \(t>0\) and the eigenvalues \(\lambda_n\) of the non-Euclidean Laplacian on \(L^2(\Gamma \setminus H)\) have the form \(\lambda_n={1\over 4}+\rho^2_n\). Previously, \textit{P. Cartier} and \textit{A. Voros} [Grothendieck Festschrift 2, Prog. Math. 87, 1-67 (1990; Zbl 0729.11024)] considered the case that \(\Gamma\setminus H\) is compact and without elliptic fixed points. Theorem 2 finds that this multi-valued theta function has a meromorphic continuation and functional equation.
congruence subgroups of the modular group
scientific article; zbMATH DE number 897672