Special points in Shimura varieties

 

December 15 – 19, 2003

 

 

Two conjectures have been made on the behavior of special points in Shimura varieties. Partial results have been obtained, but general solutions still seem far away. These conjectures were the subject of this workshop.

 

The equidistribution conjecture says that the Galois orbits of a "Hodge generic'' sequence of special points in a given Shimura variety are equidistributed for the natural hyperbolic measure. The André-Oort conjecture says that a subvariety is special if and only if it contains a Zariski-dense set of special points. As we see, these two conjectures look very similar; however these approaches, in two very different fields, are very different. On the one hand we see complex geometry and probability theory. On the other hand we see arithmetic and algebraic number theory. There are two quite different approaches to the equidistribution conjecture: the direct approach, using harmonic analysis, and the approach using rigidity arguments, via ergodic theory. All approaches were well represented at the workshop.

 

Main characteristics of this workshop:

·         people from various fields learned methods of other parts of mathematics;

·         there was an intense interaction between specialists of different fields;

·         young researchers had an active part, in the program, and in the discussions;

·         it was a pleasure to see so many aspect of modern mathematics brought together in one week;

·         maybe, an opening for a new approach to the conjectures was made;

·         the first two days were "instructional'' (on a high level); the talks in these days were coordinated beforehand; during the last three days we had more advanced talks; this set-up worked very well.

 

In the Netherlands these topics are of central interest. In three long sessions in the Intercity Number Theory Seminar introductory lectures were given as a preparation for this workshop. These lectures in November/December 2003 were pleasant, instructive, and they gave a good working ground for the Dutch participants of this workshop.

 

The meeting was attended by an audience ranging between 40 and 50 mathematicians. Among them were senior scientists with a high reputation and young PhD-students. The informal atmosphere gave the possibility for all of them to have many discussions on the topic of the workshop, but also on related topics.

 

The workshop was financed by the Lorentz Center, NWO, the Mathematical Institute of Leiden, the Leids Universiteits Fonds, MRI and the Stieltjes Instituut.

 

We thank the Lorentz Center, and especially its staff, for hospitality, for financial support, and for the cooperation with the program and various other aspects of this workshop. In preparing the workshop the assistance of the staff was very useful. The facilities and the friendly atmosphere of the Lorentz Center contributed to the level of the conference.

 

S.J. Edixhoven (University Leiden, The Netherlands)

F. Oort (University Utrecht, The Netherlands)