Abstract
We know, thanks to the work of A. Weil, that counting points of
varieties over finite fields yields purely topological information
about them. For example, an algebraic curve is topologically a
certain number g of donuts glued together. The same quantity g, on
the other hand, determines how the number of points it has over a
finite field grows as the size of this field increases.
This interaction between complex geometry, the continuous, and
finite field geometry, the discrete, has been a very fruitful two-way
street that allows the transfer of results from one context to the
other.
In this talk I will first describe how we may count the number of
points over finite fields for certain character varieties,
parameterizing representations of the fundamental group of a Riemann
surface into GL_n. The calculation involves an array of techniques
from combinatorics to the representation theory of finite groups of
Lie type. I will then discuss the geometric implications of this
computation and the conjectures it has led to.
This is joint work with T. Hausel and E. Letellier
