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).