Henri Darmon (McGill)

TITLE: Modular points on elliptic curves and cycles on Shimura varieties.

ABSTRACT. The theory of CM (Complex Multiplication) points on modular and Shimura curves provides one of the most fruitful approaches to the Birch and Swinnerton-Dyer conjecture, leading to the best known results for elliptic curves of analytic rank one. In this talk I will explore the possibility of more general constructions in which, loosely speaking, CM points on Shimura curves are replaced by higher dimensional algebraic cycles on Shimura varieties. Most of this will be a report on work in progress with Bertolini and Prasanna, and of earlier work with Charollois.