Eichler–Shimura isomorphism in higher level cases and its applications

Abstract

Let Γ be a Fuchsian group of the first kind. The Eichler–Shimura isomorphism states that the space Sk(Γ) is isomorphic to the first (parabolic) cohomology group associated to the Γ-module Rk−1 with an appropriate Γ-action. Manin reformulated the Eichler–Shimura isomorphism for the case Γ=SL2(Z) in terms of periods of cusp forms. In this paper we extend Manin's reformulation to the case Γ=Γ0+(p) with p∈{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 Γ0+(2) without using continued fractions.