Open Access
Translator Disclaimer
2012 Symmetries of the transfer operator for $\Gamma_0(N)$ and a character deformation of the Selberg zeta function for $\Gamma_0(4)$
Markus Fraczek, Dieter Mayer
Algebra Number Theory 6(3): 587-610 (2012). DOI: 10.2140/ant.2012.6.587


The transfer operator for Γ0(N) and trivial character χ0 possesses a finite group of symmetries generated by permutation matrices P with P2= 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,) of the Maass wave forms of Γ0(N). For the group Γ0(4) and Selberg’s character χα there exists just one nontrivial symmetry operator P. The eigenfunctions of the corresponding reduced transfer operator with eigenvalue λ=±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 χα.


Download Citation

Markus Fraczek. Dieter Mayer. "Symmetries of the transfer operator for $\Gamma_0(N)$ and a character deformation of the Selberg zeta function for $\Gamma_0(4)$." Algebra Number Theory 6 (3) 587 - 610, 2012.


Received: 25 January 2011; Accepted: 30 June 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1252.11068
MathSciNet: MR2966712
Digital Object Identifier: 10.2140/ant.2012.6.587

Primary: 11M36
Secondary: 11F03 , 11F72 , 35B25 , 35J05 , 37C30

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

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.6 • No. 3 • 2012
Back to Top