Shimura varieties and motives --- 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
Corrected manuscript March 13, 2018
Original manuscript February 24, 1993.

Abstract

This is the author's manuscript for
Milne, J.S., Shimura varieties and motives, In: Motives (Eds. U. Jannsen, S. Kleiman, J.-P. Serre), Proc. Symp. Pure Math., AMS, 55, 1994, Part 2, pp. 447--523.
except that the TeX has been updated, some corrections and a few minor editorial changes made, and some footnotes added. Significant changes to the original article have been noted in footnotes. The numbering is unchanged.

Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for "most" Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical model and a modular description of its points in any field containing the reflex field. Moreover, when we assume the existence of a sufficiently good theory of motives in mixed characteristic, we are able to obtain a description of the points on the Shimura variety modulo a prime of good reduction.

Contents

  1. Abelian motives and their Mumford-Tate groups
  2. Moduli of motives
  3. Shimura varieties as moduli varieties
  4. The points on a Shimura variety modulo a prime of good reduction