On Cartier-Voros type Selberg trace formula for congruence subgroups of \(\text{PSL} (2,\mathbb{R})\)
type: Item
instance of: scholarly article
title: On Cartier-Voros type Selberg trace formula for congruence subgroups of \(\text{PSL} (2,\mathbb{R})\)
zbMATH Open document ID: 0866.11037
author: Miki Hirano
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-07-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 897672
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: congruence subgroups of the modular group
zbMATH Keywords: multi-valued theta function
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
full work available at URL: pjaa.71.144
OpenAlex ID: W2085418388
cites work: Q3353097
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DOI: 10.3792/PJAA.71.144
description: scientific article; zbMATH DE number 897672