Eichler-Shimura isomorphism in higher level cases and its applications

Abstract

Let Gamma be a Fuchsian group of the first kind. The Eichler-Shimura isomorphism states that the space S-k (Gamma) is isomorphic to the first (parabolic) cohomology group associated to the Gamma-module Rk-1 with an appropriate Gamma-action. Manin reformulated the Eichler-Shimura isomorphism for the case Gamma = SL2(Z) in terms of periods of cusp forms. In this paper we extend Manin's reformulation to the case Gamma = Gamma(+)(0)(p) with p is an element of {2, 3}. The Manin relations describe relations between periods of cusp forms by using Hecke operators and continued fractions. We also extend the Manin relations and homogeneity theorem to cusp forms on Gamma(+)(0)(2) without using continued fractions. (C) 2017 Elsevier Inc. All rights reserved.