# Arithmetic of automorphic L-functions.

# Arithmetic of automorphic L-functions.

This talk will be an exposition of a circle of ideas that concerns the cohomology of arithmetic groups and the special values of certain automorphic L-functions. I will explain some recent results about the critical values of (1) Rankin-Selberg L-functions of GL(n) x GL(m) which is mostly in collaboration with Günter Harder, (2) L-functions for SO(n,n) x GL(1), in an ongoing work with Chandrasheel Bhagwat, and (3) Asai L-functions for GL(2n) over a totally real field, in an ongoing work with Muthu Krishnamurthy. Such results are obtained by studying the notion of Eisenstein cohomology. I will explain some results of Harder on the cohomology of the boundary of the Borel-Serre compactification of a locally symmetric space and its 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 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 such as in the examples listed above.