Periods of Quaternionic Shimura Varieties

In the early 80's, Shimura made a precise conjecture relating Petersson inner products of arithmetic automorphic forms on quaternion algebras over totally real fields, up to algebraic factors. This conjecture (which is a consequence of the Tate conjecture on algebraic cycles) was proved a few years later by Michael Harris. In the first half of my talk I will motivate and describe an integral version of Shimura's conjecture i.e. up to p-adic units for a good prime p . In the second half I will describe work in progress (joint with Atsushi Ichino) that makes some progress in understanding this refined conjecture.