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

Abstract

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,ℤ) 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.