Title: Computation of Zeta functions by p-adic Methods
Speaker: Ralf Gerkmann
Carslaw 273, Friday 20 August
3:05-3:55PM, 4:05-4:55PM

The aim of this talk is to give a survey on point
counting methods for varieties over finite fields
using the arithmetic of p-adic rings, with an
emphasis on Kedlaya's algorithm and its recent
extensions. After a short introduction on zeta
functions in general, in the first part I describe
different algorithmic approaches together with their
underlying cohomology theory: Satoh's algorithm,
Mestre's AGM method and the work due to Lauder and
Wan based on Dwork's p-adic analysis.
The second part concentrates on rigid cohomology,
starting with a short review of its construction.
Then it is explained how rigid cohomology is used for the
computation of zeta functions. Above all, I will focus
on the problems that occur in generalizing the particular
steps of Kedlaya's algorithm to more general varieties.
Finally I will introduce the notion of F-isocrystals
(the natural coefficients in rigid cohomology) and
describe possible applications related to point
counting.