The p > 2 theory of Shimura varieties

Adrian Vasiu, Univ. of Utah Abstract A central problem in mathematics is to classify different objects. The problems

can be of discrete nature, or of continuous nature. One of the main achievements of classical Hodge theory is in showing that inside many classes of polarized, connected, projective, smooth varieties over the field of complex numbers endowed with some extra structure, the isomorphism classes are in one-to-one correspondence with the complex points of an open subvariety U of some varieties obtained by taking quotients of Hermitian symmetric domains, by arithmetic subgroups. We motivate briefly why it is desirable to have good integral models of U over the ring of integers of suitable number fields, punctured in some points. The existence of such good models, in the context of Shimura varieties, was hinted at by Langlands, and precisely conjectured by Milne. In this talk and the one of the next week, we will report on three fundamental theorems pertaining to these integral models, provided we restrict to the class of Shimura varieties of preabelian type (i.e. to the maximal class which can be studied via abelian varieties). So in the first talk we refer just to this class, and just to the first two theorems.

First, we proved this conjecture for primes p > 2. Second, we completely classified the good morphisms between them. There is no a priori knowledge required. We will assume just elementary prerequisite on reductive groups. The proofs rely on fundamental works of Shimura, Satake, Deligne, Mumford, Milne, Faltings, de Jong and the author, which will be reviewed to some extend during the talk.

These integral models are the central objects of study of many parts of mathematics, ranging from number theory, Galois representation, Hodge theory, diophantine geometry, representation theory, etc.