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

TL;DR

This paper investigates symmetries of transfer operators for congruence subgroups, relating them to automorphisms of Maass forms, and explores how these symmetries influence the zeros of associated Selberg zeta functions under character deformations.

Contribution

It identifies specific symmetry operators for transfer operators of \\Gamma_0(N) and analyzes their impact on the factorization of Selberg zeta functions and the behavior of zeros under character deformation.

Findings

Symmetries lead to factorization of Selberg zeta functions.

Eigenfunctions relate to Maass forms with automorphism symmetry.

Zeros of the Selberg function behave predictably under deformation, staying on or leaving the critical line.

Abstract

The transfer operator for $\Gamma_0(N)$ and trivial character $\chi_0$ 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 $GL(2,\mathbb{Z})$ of the Maass wave forms of $\Gamma_0(N)$ . For the group $\Gamma_0(4)$ and Selberg's character $\chi_\alpha$ there exists just one non-trivial symmetry operator $P$. The eigenfunctions of the corresponding reduced transfer operator with eigenvalue $\lambda=\pm 1$ are related to Maass forms even respectively odd under a corresponding automorphism. It then follows from a result of Sarnak and Phillips that the zeros of the Selberg function determined by the eigenvalues $\lambda=-1$ of the reduced transfer operator stay on the critical line under the deformation of the character. From numerical results we expect that on the other hand all the zeros corresponding to the eigenvalue $\lambda=+1$ leave this line for $\alpha$ turning away from zero.