Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 10 November 2003, 16:00 - 17:00, room 402

Dmitry Logachev (Universidad Simon Bolivar, Venezuela) Action of Hecke correspondences on subvarieties of Shimura varieties.

Abstract: Let X be an irreducible component of a Shimura variety of a fixed level, T a p-Hecke correspondence on X, V an irreducible component of a Shimura subvariety of X. So, T(V) is a union of irreducible components of some other Shimura subvarieties of X.

There are 2 natural problems:

1. To describe the structure of T(V), particularly its representation as a union of Shimura subvarieties, their irreducible components and Galois action on them.

2. To study their reduction at p.

If X is a curve then solutions of both these problems are used in Kolyvagin's proof of finiteness of SH of elliptic curves over Q. We can hope that the solution of these problems in the general case will be used for a generalization of these results to the case of submotives of Shimura varieties.

In the present talk we give an (almost complete) solution of these problems for some simple types of X, V. As a result, we get an evidence that for (almost) all cases the first problem can be solved (which is not clear beforehand), and the result can be simply formulated in terms of geometry of T(t) (which is a finite scheme over Spec (Z/p^n), n depends on T). Results for different types of X and T are quite different and non-expected.

Remark. The technique of the solution of the first problem is "elementary" (matrix calculations). Since only few types of X, T, V are investigated, these problems represent a large field of further activity. In a process of solution we get some non-trivial algebraic implications: one set of polynomial equalities implies another one, i.e. we have ideals I, J in a ring of polynomials such that I is a subset of Rad(J). In a few cases where answers are not complete, it is possible to use computer calculations on order to state conjectures and to check them for some small p.