**Description and Aim**

** **

**Stieltjes**** Week**

This DIAMANT Stieltjesweek aims to provide an introduction to three hot
topics in arithmetic geometry: L-series, the Birch and Swinnerton-Dyer
conjecture, and the Batyrev-Manin conjecture. Through
three mini-courses the audience will be prepared for the conference Counting
points on varieties to be held during the following week.

Each morning
will feature three lectures, one for each mini-course. In the afternoon there
will be problem sessions. See the program
for details.

Ted Chinburg (University
of Pennsylvania) will
lecture on L-series. A key step in the study of L-series is to relate these to cohomology theories, such as étale
cohomology. More recently, other cohomology
theories such as Weil-étale cohomology,
Arakelov Chow groups and the foliation cohomology associated to dynamical systems have been used
to study L-series. One goal of the workshop will be to give an overview of both
classical work and recent developments concerning relations between L-series
and cohomology.

Tim Dokchitser (University of Cambridge) will give an
introduction to the Birch and Swinnerton-Dyer
conjecture, which relates the rank of an elliptic curve *E* over the rational numbers to the behavior of its L-function *L( E, s )* near *s = 1*. This L-function encodes the number of points of *E* modulo *p* for all prime numbers *p*.

Ronald van Luijk (Universiteit Leiden) will
lecture on the Batyrev-Manin conjecture. For any
variety *X* over the rational numbers
and any real number *B*, one can
consider the rational points on *X* for
which the absolute value of the coordinates have numerator and denominator bounded
by *B*. The Batyrev-Manin
conjecture predicts the asymptotic growth of the number of these points as a
function of *B* for interesting classes
of varieties *X*.

The lectures
are aimed at Master's and PhD students, as well as young postdocs,
in arithmetic geometry and related areas. Familiarity with the basic theorems
of algebraic number theory and algebraic geometry will be assumed.