The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by Shimura (1959) for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms.
There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in (Gunning 1961).
There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of real cohomology, which relates the coefficients of cusp forms to eigenvalues of Frobenius acting on these groups.
If G is a Fuchsian group and M is a representation of it then the Eichler cohomology group H1P(G,M) is defined to be the kernel of the map from H1(G,M) to Πc H1(Gc,M), where the product is over the cusps c of a fundamental domain of G, and Gc is the subgroup fixing the cusp c. The Eichler–Shimura isomorphism is an isomorphism between the space of cusp forms on G of weight n + 2 and the first Eichler cohomology of the group G with the coefficients in the G-module
depends on n (Shimura, "Intruduction to the arithmetic theory of automorphic functions", Theorem 8.4)