Shimura Varieties and the Work of Langlands. J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate
Errata

pdf (current version 1.00).

Abstract

Shimura varieties are initially defined as complex manifolds (quotients of Hermitian symmetric domains by congruence groups), but is known that they are algebraic varieties defined in a natural way over number fields. The oldest examples are the elliptic modular curves (quotients of the complex upper half plane by congruence subgroups of SL(2,Z). In the talk, I'll explain these two sentences, and also why Langlands was so interested in Shimura varieties.

About

These are my notes for a "popular" talk in the `What is ...?' seminar at the University of Michigan, November 17, 2010.

History

v1.00 (November 18, 2009). First version on the web (11 pages).