Abstract
In this paper we discuss the transfer operator approach to Selberg’s zeta function for P S L(2, ℤ). Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.
The work is supported by DFG Schwerpunktprogramm “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme”.
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References
E. Artin, Ein mechanisches System mit quasiergodischen Bahnen, Collected papers, Addison-Wesley, Reading MA. (1965), pp. 499–504.
R. Adler, L. Flatto, Cross section maps for the geodesic now on the modular surface, Contemp. Math. 26, Am. Math. Soc. Providence RI. (1984), pp. 9–24.
P. Cartier, An introduction to zeta functions, in “From Number theory to Physics” eds: M. Waldschmidt et al., Springer Verlag, Berlin (1992).
B. Dwork, On the rationality of the zeta function of an algebraic variety, Am. J. Math. 82 (1960), pp. 631–648.
P. Deligne, La conjecture de Weil I, Publ. Math. IHES, no. 43. Press Univ. France (1974).
M. Gutzwiller, Chaos in Classical and Quantum mechanics, Springer-Verlag, New York (1990).
A. Grothendieck, Formule de Lefschetz et rationalité de fonctions L, Sem. Bourbaki no. 279, Benjamin, N.Y. (1966).
A. Grothendieck, La théorie de Fredholm, Bull. Soc. math. France 84 (1956), pp. 319–384.
K. Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, Inc. (1965).
J. Lewis, Spaces of Holomorphic Functions Equivalent to the Even Maass Cusp Forms, Invent. Math. 127 (1997), pp. 271–306.
J. Lewis, private communication.
A. Lasota, M. Mackey, Probabilistic properties of deterministic systems, Cambridge Univ. Press, Cambridge (1985).
J. Lewis, D. ZAGIER, Period functions and the Selberg zeta function for the modular group, Preprint MPI Mathematik Bonn, MPI 96/112 (1996).
A. Manning, Dynamics of geodesic and horocycle rows on surfaces of constant negative curvature, in “Ergodic theory, symbolic dynamics and hyperbolic spaces” eds: T. Bedford et al., Oxford Univ. Press (1991), pp. 71–91.
D.H. Mayer, Continued fractions and related transformations, in “Ergodic theory, symbolic dynamics and hyperbolic spaces” eds: T. Bedford et al., Oxford Univ. Press (1991), pp. 175–222.
D.H. Mayer, On the Thermodynamic Formalism for the Gauss Map, Commun. Math. Phys. 130 (1990), pp. 311–333.
D.H. Mayer, The thermodynamic formalism approach to Selberg’s zeta function for P S L (2, ℤ), Bull. Am. Math. Soc. 25 (1991), pp. 55–60.
W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin (1966).
N. Nielsen, Traité Élémentaire des Nombres de Bernoulli, Gauthier-Villars, Paris (1923).
P. Robba, Une introduction naive aux cohomologies de Dwork, Soc. Math. France 2e Serie, Memoire no. 23 (1986), pp. 61–105.
D. Ruelle, Zeta functions and statistical mechanics, Soc. Math. France. Asterisque 40. (1976), pp. 167–176.
D. Ruelle, Generalized zeta functions for Axiom A basic sets, Bull. Am. Math. Soc. 82 (1976), pp. 153–156.
D. Ruelle, Zeta functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), pp. 231–242.
D. Ruelle, Thermodynamic formalism, Addison-Wesley, Reading MA (1978).
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Ind. Math. Soc. 20 (1956), pp. 47–87.
P. Sarnak, Arithmetic Quantum Chaos, Schur lectures Tel Aviv 1992; Israel math. conf. proceedings 8 (1995), pp. 183–236.
C. Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), pp. 69–80.
D. Zagier, Periods of modular forms, traces of Hecke operators and multiple zeta values, in “Studies of automorphic forms and L-functions” RIMS Kyoto (1992), pp. 162–170.
D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991), pp. 449–465.
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Chang, CH., Mayer, D.H. (1999). The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ). In: Hejhal, D.A., Friedman, J., Gutzwiller, M.C., Odlyzko, A.M. (eds) Emerging Applications of Number Theory. The IMA Volumes in Mathematics and its Applications, vol 109. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1544-8_3
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