ncts

Let f be a cohomological cuspidal modular form and F its base change to some quadratic field E. In the late 90s, Hida has established a p-adic divisibility between the twisted adjoint L-value of f (normalized by some period) and some congruence number controlling the congruences between F and non-base-change forms on E. This divisibility has later been proved to be an equality by Tilouine and Urban, and used by them to establish some p-adic integral relation between the periods of f and F. For general GL(n), in addition to the Arthur-Clozel base change, there is the Rogawski-Mok stable base change from the quasi-split unitary group associated to E. More problematically, in this context, the very definition of the periods poses a problem because the dimension of the middle degree cuspidal cohomology groups is too large. In this talk, I will explain how however for GL3 over a real quadratic field, one can define appropriate periods associated to base changes and to what extent it is possible to extend the results of Hida for both base changes. If time allows, I will also explain how to use these results to establish some p-adic divisibilities between these middle degree periods and top/bottom degree periods. This is a work in progress.