Symmetries of the transfer operator for \(\Gamma_0(N)\) and a character deformation of the Selberg zeta function for \(\Gamma_0(4)\)
type: Item
instance of: scholarly article
title: Symmetries of the transfer operator for \(\Gamma_0(N)\) and a character deformation of the Selberg zeta function for \(\Gamma_0(4)\)
zbMATH Open document ID: 1252.11068
published in: Algebra \& Number Theory
publication date: 2012-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.4441
full work available at URL: p09.xhtml
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 37C30
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 6064723
zbMATH Keywords: transfer operator
zbMATH Keywords: factorization of the Selberg zeta function
zbMATH Keywords: Hecke congruence subgroups
zbMATH Keywords: Maass wave forms
zbMATH Keywords: character deformation
author: Markus Fraczek
author: Dieter Mayer
MaRDI profile type: Publication
OpenAlex ID: W2963773897
arXiv ID: 1011.4441
DOI: 10.2140/ANT.2012.6.587
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review text: Let \(\Gamma=\Gamma_0(N) \subset \mathrm{SL}(2,\mathbb{Z})\) denote the Hecke congruence subgroup of level \(N\) containing all matrices of the form \({ a \;\;\; b \choose Nc \;\; d }\) with integer entries and determinant \(1\). Its index is \(\mu_N= N \prod_{p\mid N}(1+1/p)\) with \(p\) running through the prime-divisors of \(N\). \(\chi\) denotes a unitary character of \(\Gamma\).NEWLINENEWLINEWe consider the geodesic flow \(\Phi_t: SM_\Gamma \to SM_\Gamma\) on the unit tangent bundle \(SM_\Gamma\) of the corresponding modular surface \(M_\Gamma = \Gamma \backslash \mathbb{H}\) of constant negative curvature and \(\mathbb{H}\) denoting the upper complex half-plane.NEWLINENEWLINELet \(\rho:\Gamma \to \text{end}(\mathbb{C}^d)\) denote a unitary representation of \(\Gamma\). The Selberg zeta function \(Z_{\Gamma,\rho}\) is defined as NEWLINE\[NEWLINE Z_{\Gamma,\rho}(\beta) = \prod_\gamma \prod_{k=0}^\infty \det\bigg(1-\rho(g_\gamma) \, \text{exp}\big(-(k+\beta)l_\gamma) \big) \bigg) NEWLINE\]NEWLINE where \(l_\gamma\) denotes the period (i.e.\ length) of the prime periodic orbit \( \gamma\) of \(\Phi_t\) and \(g_\gamma \in \Gamma\) is hyperbolic with \(g_\gamma(\gamma)=\gamma\).NEWLINENEWLINEThe dynamical approach is to express the Selberg zeta function in terms of the so-called transfer operator, known from Ruelle's thermodynamic formalism approach to dynamical systems. The transfer operator \(L_{\beta,\chi}: B \to B\) was determined in [\textit{C.-H.\ Chang} and \textit{D.\ Mayer}, in: Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, 523--562 (2001; Zbl 1211.37030)] and [\textit{C.-H.\ Chang} and \textit{D.\ Mayer}, Regul.\ Chaotic Dyn.\ 5, No.\ 3, 281--312 (2000; Zbl 0979.37014)] as NEWLINE\[NEWLINE L_{\beta,\chi} = { 0 \;\; \mathcal{L}^+_{\beta, \rho_\chi} \choose \mathcal{L}^-_{\beta, \rho_\chi} \;\; 0 } NEWLINE\]NEWLINE where \(\chi\) is a unitary character of \(\Gamma\), \(B=B(D,\mathbb{C}^\mu) \oplus B(D,\mathbb{C}^\mu)\) is…
description: scientific article; zbMATH DE number 6064723