Introduction

The purpose of this paper is to review the strategies and methods developed over the past few years in relation to the André-Oort conjecture. We would like bring the attention of the reader to a survey by Rutger Noot (see [20]) based on his talk at the Seminaire Bourbaki in November 2004. In this paper, we have tried to avoid overlapping too much with Noot's survey. This paper is based on the author's talk at the Durham Symposium in the summer of 2004. Laurent Clozel gave a talk on an approach to the Andre-Oort conjecture involving ergodic-theoretic methods. We will touch upon the contents of his lecture in the last section of this paper.

Let us recall the statement of the Andre-Oort conjecture.

Conjecture 1.1 (André-Oort)Let S be a Shimura variety and let Σ be a set of special points in S. The irreducible components of the Zariski closure of Σ are special subvarieties (or subvarieties of Hodge type).

In the next section we will review the notions of Shimura varieties, special points and special subvarieties. In this introduction we review some of the results on this conjecture obtained so far. This conjecture was stated by Yves André in 1989 (Problem 9 in [1]) for one dimensional subvarieties of Shimura varieties and in 1995 by Frans Oort in [22] for subvarieties of the moduli space Ag of principally polarised abelian varieties of dimension g. The statement above is the obvious generalisation of these two conjectures and is now refered to as the André-Oort conjecture.