Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)

Fraczek, Markus; Mayer, Dieter

doi:10.2140/ant.2012.6.587

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Symmetries of the transfer operator for $\Gamma_0(N)$ and a character deformation of the Selberg zeta function for $\Gamma_0(4)$

Markus Fraczek and Dieter Mayer

Vol. 6 (2012), No. 3, 587–610

DOI: 10.2140/ant.2012.6.587

Abstract

The transfer operator for $Γ_{0} (N)$ and trivial character $χ_{0}$ possesses a finite group of symmetries generated by permutation matrices $P$ with $P^{2} = id$ . Every such symmetry leads to a factorization of the Selberg zeta function in terms of Fredholm determinants of a reduced transfer operator. These symmetries are related to the group of automorphisms in $GL (2, ℤ)$ of the Maass wave forms of $Γ_{0} (N)$ . For the group $Γ_{0} (4)$ and Selberg’s character $χ_{α}$ there exists just one nontrivial symmetry operator $P$ . The eigenfunctions of the corresponding reduced transfer operator with eigenvalue $λ = \pm 1$ are related to Maass forms that are even or odd, respectively, under a corresponding automorphism. It then follows from a result of Sarnak and Phillips that the zeros of the Selberg function determined by the eigenvalue $λ = - 1$ of the reduced transfer operator stay on the critical line under deformation of the character. From numerical results we expect that, on the other hand, all the zeros corresponding to the eigenvalue $λ = + 1$ are off this line for a nontrivial character $χ_{α}$ .

Keywords

transfer operator, Hecke congruence subgroups, Maass wave forms, character deformation, factorization of the Selberg zeta function

Mathematical Subject Classification 2010

Primary: 11M36

Secondary: 35J05, 35B25, 37C30, 11F72, 11F03

Milestones

Received: 25 January 2011

Accepted: 30 June 2011

Published: 5 July 2012

Authors

Markus Fraczek
Institut für Theoretische Physik Technische Universität Clausthal Leibnizstraße 10 D-38678 Clausthal-Zellerfeld Germany

Dieter Mayer
Institut für Theoretische Physik Lower Saxony Professorship Technische Universität Clausthal Leibnizstraße 10 D-38678 Clausthal-Zellerfeld Germany

Vol. 6, No. 3, 2012