Period Functions and the Selberg Zeta Function for the Modular Group

The Selberg trace formula on a Riemann surface X connects the discrete spectrum of the Laplacian with the length spectrum of the surface, that is, the set of lengths of the closed geodesics of on X. The connection is most strikingly expressed in terms of the Selberg zeta function, which is a meromorphic function of a complex variable s that is defined for <(s) > 1 in terms of the length spectrum and that has zeros at all s ∈ C for which s(1 − s) is an eigenvalue of the Laplacian in L(X). We will be interested in the case when X is the quotient of the upper half-plane H by either the modular group Γ1 = SL(2,Z) or the extended modular group Γ = GL(2,Z), where γ = ( a b c d ) ∈ Γ acts on H by z 7→ (az + b)/(cz + d) if det(γ) = +1 and z 7→ (az̄ + b)/(cz̄ + d) if det(γ) = −1. In this case the length spectrum of X is given in terms of class numbers and units of orders in real quadratic fields, while the eigenfunctions of the Laplace operator are the non-holomorphic modular functions usually called Maass wave forms. (Good expositions of this subject can be found in [6] and [7]).