In
mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
s, introduced by , that is a variation of group cohomology analogous to the image of the
cohomology with compact support In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
Singular cohomology with compact support
Let X be a topological space. Then
:\d ...
in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by 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 real cohomology, which relates the coefficients of cusp forms to eigenvalues of
Frobenius acting on these groups. used this to reduce the
Ramanujan conjecture to the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
that he later proved.
Eichler cohomology
If ''G'' is a
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
and ''M'' is a representation of it then the Eichler cohomology group ''H''(''G'',''M'') is defined to be the kernel of the map from ''H''(''G'',''M'') to
Π
''c'' ''H''(''G''
''c'',''M''), where the product is over the cusps ''c'' of a fundamental domain of ''G'', and ''G''
''c'' is the subgroup fixing the cusp ''c''.
References
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{{DEFAULTSORT:Eichler-Shimura isomorphism
Modular forms