Rational period functions of the modular group II

Abstract

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL (2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q ( z ) which occur in functional equations of the form where k ∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F ( z ) at ∞: with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q ( z ) the period of F .