REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The Langlands program gives conjectural relations between representations of the Galois group and modular forms. For instance, Taniyama-Shimura conjecture, "Every elliptic curve is modular", was a central part of Wiles' proof of Fermat's Last Theorem. Modular forms transform via the action of SL(2,Z) or its congruence subgroups. Most of the time, modular forms can be described cohomologically. In this talk we present computations of cohomology of GL(3,Z) and GL(4,Z) and some of their congruence subgroups with coefficients in finite dimensional highest weight representations. The results for GL(3,Z) are joint work with Bajpai, Harder and Moya Giusti. Some of the computation of cohomology of GL(4,Z) is published in a solo paper. The rest of the computations for GL(4,Z) and for some congruence subgroups are part of a book in preparation (180 pages so far).