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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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,) 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

Source
Algebra Number Theory, Volume 6, Number 3 (2012), 587-610.

Dates
Received: 25 January 2011
Accepted: 30 June 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729805

Digital Object Identifier
doi:10.2140/ant.2012.6.587

Mathematical Reviews number (MathSciNet)
MR2966712

Zentralblatt MATH identifier
1252.11068

Subjects
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

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

Citation

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


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References

  • R. W. Bruggeman, M. Fraczek, and D. Mayer, “Perturbation of zeros of the Selberg zeta function for $\Gamma_0(4)$”, preprint, 2012.
  • C.-H. Chang and D. Mayer, “The period function of the nonholomorphic Eisenstein series for ${\rm PSL}(2,{\bf Z})$”, Math. Phys. Electron. J. 4 (1998), no. 6.
  • C.-H. Chang and D. Mayer, “Thermodynamic formalism and Selberg's zeta function for modular groups”, Regul. Chaotic Dyn. 5:3 (2000), 281–312.
  • C.-H. Chang and D. H. Mayer, “Eigenfunctions of the transfer operators and the period functions for modular groups”, pp. 1–40 in Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999), edited by M. L. Lapidus and M. van Frankenhuysen, Contemp. Math. 290, Amer. Math. Soc., Providence, RI, 2001.
  • C.-H. Chang and D. H. Mayer, “An extension of the thermodynamic formalism approach to Selberg's zeta function for general modular groups”, pp. 523–562 in Ergodic theory, analysis, and efficient simulation of dynamical systems, edited by B. Fiedler, Springer, Berlin, 2001.
  • J. H. Conway and S. P. Norton, “Monstrous moonshine”, Bull. London Math. Soc. 11:3 (1979), 308–339.
  • A. Deitmar and J. Hilgert, “A Lewis correspondence for submodular groups”, Forum Math. 19:6 (2007), 1075–1099.
  • I. Efrat, “Dynamics of the continued fraction map and the spectral theory of ${\rm SL}(2,\mathbb Z)$”, Invent. Math. 114:1 (1993), 207–218.
  • M. Fraczek, Character deformation of the Selberg zeta function for congruence subgroups via the transfer operator, thesis, Clausthal Institute of Technology, 2010.
  • D. A. Hejhal, The Selberg trace formula for ${\rm PSL}(2,\,{\bf R})$, vol. 2, Lecture Notes in Mathematics 1001, Springer, Berlin, 1983.
  • J. Lehner and M. Newman, “Weierstrass points of $\Gamma _{0}\,(n)$”, Ann. of Math. (2) 79 (1964), 360–368.
  • J. Lewis and D. Zagier, “Period functions for Maass wave forms, I”, Ann. of Math. (2) 153:1 (2001), 191–258.
  • Y. I. Manin and M. Marcolli, “Continued fractions, modular symbols, and noncommutative geometry”, Selecta Math. (N.S.) 8:3 (2002), 475–521.
  • D. Mayer and T. Mühlenbruch, “Nearest $\lambda_q$-multiple fractions”, pp. 147–184 in Spectrum and dynamics, edited by D. Jakobson et al., CRM Proc. Lecture Notes 52, Amer. Math. Soc., Providence, RI, 2010.
  • D. Mayer and F. Str ömberg, “Symbolic dynamics for the geodesic flow on Hecke surfaces”, J. Mod. Dyn. 2:4 (2008), 581–627.
  • D. Mayer, T. Mühlenbruch, and F. Strömberg, “The transfer operator for the Hecke triangle groups”, Discrete Contin. Dyn. Syst. A 32:7 (2012), 2453–2484.
  • T. Mühlenbruch, “Hecke operators on period functions for $\Gamma\sb 0(n)$”, J. Number Theory 118:2 (2006), 208–235.
  • R. Phillips and P. Sarnak, “Cusp forms for character varieties”, Geom. Funct. Anal. 4:1 (1994), 93–118.
  • A. Selberg, “Remarks on the distribution of poles of Eisenstein series”, pp. 251–278 in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part 2 (Ramat Aviv, 1989), edited by S. Gelbart et al., Israel Math. Conf. Proc. 3, Weizmann, Jerusalem, 1990.
  • F. Strömberg, “Computation of Selberg's zeta functions on Hecke triangle groups”, preprint, 2008.
  • A. Venkov, “The theory of the Selberg zeta-function”, pp. 47–54 in Spectral theory of automorphic functions and its applications, Math. Appl. (Soviet Ser.) 51, Kluwer, Dordrecht, 1990.
  • D. Zagier, “New points of view on the Selberg zeta function”, in Proceedings of Japanese–German Seminar: explicit structures of modular forms and zeta functions (Hakuba, 2001), edited by T. Ibukiyama, Ryushi-Do, Nagano, 2002.