Short course "Eisenstein Cohomology and Special Values of L-functions". Lecture-2: Special values of automorphic L-functions

Short course 2/3, online talk.

The second talk will be a continuation of the first talk, where I will discuss an idea of Harder's which one sees in his early papers from the 70's and 80's on arithmetic applications of Eisenstein cohomology (especially, his pioneering 1987 article in Inventiones on GL(2)). This idea gives a cohomological interpretation of Langlands's constant term theorem and offers the hope of proving rationality results for the critical values of L-functions. Such a program was brought to fruition for Rankin-Selberg L-functions for GL(n) x GL(m) over a totally real number field. Subsequently, I have proved analogous theorems for some other Langlands-Shahidi L-functions, such as the Rankin-Selberg L-functions over a totally imaginary base field, for L-functions for O(2n) over a totally real field with Chandrasheel Bhagwat, and in an ongoing project with Muthu Krishnamurthy for Asai L-functions for GL(n) and a quadratic extension of totally real fields. In the second talk, I will discuss how one proves these results on L-values and especially what are the challenges one must overcome in these special cases.