Canonical models of (mixed) Shimura varieties and automorphic vector bundles. (English) Zbl 0704.14016

Milne, J. S.

Automorphic forms, Shimura varieties, and L-functions. Vol. I, Proc. Conf., Ann Arbor/MI (USA) 1988, Perspect. Math. 10, 283-414 (1990).

[For the entire collection see Zbl 0684.00003.]
The author reviews the theory of canonical models of Shimura-varieties and related objects. That is a Shimura-variety is a priori defined over the complex numbers, and one seeks a model over a number-field. Even better one can show that any conjugate of a Shimura-variety is again such a variety. The authors gives some proofs for the simple results, so that he can explain what are the more difficult ones. For these he explains only the strategy, but not the details.
All in all an excellent overview of the theory.

Reviewer: G.Faltings

MSC:

14G35	Modular and Shimura varieties
11G18	Arithmetic aspects of modular and Shimura varieties
14A20	Generalizations (algebraic spaces, stacks)
11F99	Discontinuous groups and automorphic forms

Keywords:

Tannakian categories; Hodge cycles; motives; Taniyama group; canonical models; automorphic vector bundles; canonical models of Shimura- varieties; model over a number-field

Citations:

Zbl 0684.00003