A. Raghuram (Purdue U.)

Eisenstein cohomology and the special values of automorphic L-functions.

Abstract: This talk will be an exposition of a circle of ideas that concerns the cohomology of arithmetic groups and the special values of automorphic L-functions. I will begin by introducing the general context in which one can study the notion of Eisenstein cohomology. I will then explain some results of Harder on the cohomology of the boundary of the Borel-Serre compactification of a locally symmetric space and it's relation with induced representations of the ambient reductive group. Once this context is in place one may then try to view Langlands's constant term theorem, which sees the ratios of products of automorphic L-functions, in terms of maps in cohomology. Whenever this is possible one is able to prove rationality results for ratios of critical values of certain automorphic L-functions. I will explain some recent results about (1) Rankin-Selberg L-functions of GL(n) x GL(m)--mostly in collaboration with Gunter Harder, and (2) L-functions for SO(n,n) x GL(1)--in collaboration with Chandrasheel Bhagwat.