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

Abstract

The transfer operator for ${\Gamma }_0\left(N\right)$ 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 $GL\left(2,\mathbb{Z}\right)$ of the Maass wave forms of ${\Gamma }_0\left(N\right)$. For the group ${\Gamma }_0\left(4\right)$ and Selberg’s character ${\chi }_{\alpha }$ there exists just one nontrivial symmetry operator $P$. The eigenfunctions of the corresponding reduced transfer operator with eigenvalue $\lambda =\pm 1$ 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 $\lambda =-1$ 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 $\lambda =+1$ are off this line for a nontrivial character ${\chi }_{\alpha }$.