Speaker: Kazuhiro Fujiwara (Nagoya)

Title: Arithmetic geometry and Shimura varieties

Abstract: In 1960's, new ideas and techniques were born in abstract algebraic geometry under a strong influence of A. Grothendieck. Because of it's generality, this new style of geometry -arithmetic geometry- has put much impact on number theory, which were already dreamed of by L. Kronecker in the 19th century. These new tools, especially new cohomology theories for algebraic varieties, have offered strong methods to study varieties which were already known to be important. From the viewpoint of number theory, Shimura varieties, which are arithmetic quotients of hermitian symmetric spaces, form such a class.

In this lecture, I will try to explain how general theories in arithmetic geometry can be applied to study Shimura varieties, and how Shimura varieties offer a good class of examples for general theories such as Lefschetz-Verdier trace formulas, rigid-analytic geometry and the geometry of positive characteristics. A focus will be put on explicit mysterious identities (non-abelian reciprocity law).