In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by Eichler , that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by Shimura 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 . There is also a variation of the Eichler–Shimura isomorphisms using l-adic cohomology instead of re