Shimura varieties are the natural generalization of elliptic modular curves. Examples include the Hilbert modular varieties and the Siegel modular varieties. The fundamental theorem in the theory of Shimura varieties is the existence and uniqueness of canonical models over number fields. A primary goal of the course will be to obtain a good understanding of this theorem. In particular, we shall discuss the theorem of Shimura and Taniyama on complex multiplication, and the various ways of realizing Shimura varieties as moduli varieties. We expect also to include the following two topics: the structure of Shimura varieties modulo p, especially in the PEL case; boundaries of Shimura varieties and their various compactifications.

Slides: Introduction to Shimura Varieties (.pdf) (.ps)