#### May 13 – 18, 2006

**Organizers**: **Andrew Granville** (Montréal),** **

Yuri Tschinkel (Göttingen),

**Michael Bennett **(UBC), **Chantal David** (Concordia)

et** Bill Duke** (UCLA).

This workshop will include expository talks as well as presentations on current research highlighting the flow of ideas between analytic number theory and arithmetic geometry. On the analytic side, one has the circle method and its modern adaptations, sieving methods, techniques from spectral theory, ergodic theory and the theory of automorphic forms. On the side of arithmetic geometry, there is the theory of universal torsors, heights, intersection theory, p-adic integration, theory of moduli spaces, compactifications of algebraic groups and homogeneous spaces. Open problems range from understanding very concrete equations to more abstract questions of proving and interpreting asymptotics of rational and integral points on orbits of linear algebraic groups, special points on semi-abelian varieties and Shimura varieties.

**Among the themes to be discussed are**:

- Theory of height functions;
- Analytic approaches to the existence of rational and integral points on algebraic varieties;
- Counting of points of bounded heights (e.g., asymptotics for Fano varieties); à
- Equidistribution theorems;

- Geometric techniques in analytic and algebraic number theory (e.g., symmetric products, fibration methods, averaging in families);
- Automorphic forms and distribution of special points on Shimura varieties.

This meeting brings together the participants of the MSRI and CRM workshops.The meeting will be held at the

Banff International Research Station.