Introduction to Shimura Varieties. 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
pdf (current version 23.10.04).


This article is an introduction to the theory of Shimura varieties or, in other words, the arithmetic theory of automorphic functions and holomorphic automorphic forms.


Hermitian symmetric domains; Hodge structures and their classifying spaces; locally symmetric varieties; connected Shimura varieties; Shimura varieties; the Siegel modular variety; Shimura varieties of Hodge type; PEL Shimura varieties; general Shimura varieties; complex multiplication (the Shimura-Taniyama formula and the main theorem); definition of canonical models; uniqueness of canonical models; existence of canonical models; abelian varieties over finite fields; the good reduction of Shimura varieties; a formula for the number of points.


These are my notes for a series of 15 lectures at the Clay summer school, 2003. They contain footnotes and endnotes not in the version published in: Harmonic Analysis, the Trace Formula and Shimura Varieties (James Arthur, Robert Kottwitz, Editors) AMS, 2005, (Lectures at the Summer School held at the Fields Institute, June 2 -- June 27, 2003).