Algebra & Number Theory
- Algebra Number Theory
- Volume 6, Number 3 (2012), 587-610.
Symmetries of the transfer operator for $\Gamma_0(N)$ and a character deformation of the Selberg zeta function for $\Gamma_0(4)$
The transfer operator for and trivial character possesses a finite group of symmetries generated by permutation matrices with . 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 of the Maass wave forms of . For the group and Selberg’s character there exists just one nontrivial symmetry operator . The eigenfunctions of the corresponding reduced transfer operator with eigenvalue 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 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 are off this line for a nontrivial character .
Algebra Number Theory, Volume 6, Number 3 (2012), 587-610.
Received: 25 January 2011
Accepted: 30 June 2011
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx] 35B25: Singular perturbations 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 11F72: Spectral theory; Selberg trace formula 11F03: Modular and automorphic functions
Fraczek, Markus; Mayer, Dieter. 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 (2012), no. 3, 587--610. doi:10.2140/ant.2012.6.587. https://projecteuclid.org/euclid.ant/1513729805