Introduction to Shimura Varieties. J.S. Milne Top

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 23.10.04).
### Abstract

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.
### Contents

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.
### About

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).