Algebra & Number Theory

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

Markus Fraczek and Dieter Mayer

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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.

Article information

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

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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

transfer operator Hecke congruence subgroups Maass wave forms character deformation factorization of the Selberg zeta function


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.

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