**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)