Shimura Varieties and Moduli. 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 file for my manuscript (30.04.11, 76 pages).
pdf file for published version (15.10.12, 82 pages).
Caution: the numbering differs (see below).


Connected Shimura varieties are the quotients of hermitian symmetric domains by discrete groups defined by congruence conditions. We examine their relation with moduli varieties. As much as possible, I have included complete proofs.


Elliptic modular curves; hermitian symmetric domains; discrete subgroups of Lie groups; locally symmetric varieties; variations of Hodge structures; Mumford-Tate groups and their variation in families; Period subdomains; variations of Hodge structures on locally symmetric varieties; absolute Hodge classes and motives; symplectic representations; moduli.

Published version

The article was published in Handbook of Moduli (Gavril Farkas, Ian Morrison, Editors), International Press of Boston, 2013, Vol II, 462--544.
In the published version, equations and theorem-like statements are numbered in sequence. For example, equation (10), p25, has become (6.2) (first equation after 6.1), and Proposition 6.2 has become 6.3 (third numbered equation or theorem-like statement in section 6).


v1.00 (December 1, 2010) First version on the web. 69 pages. pdf
v2.00 (April 30, 2011). Improvements to the exposition. some misprints fixed.