Ralf Gerkmann

Thursday 22th March, 4:05-4:55PM, Carslaw 375

Relative rigid cohomology and point counting on hypersurfaces

In 2003 Alan Lauder presented an algorithm, based on Dwork's p-adic homology theory, which computes the zeta function of a hypersurface over a finite field by considering its deformation into another one with especially simple structure (a diagonal hypersurface). We translate this approach into the language of rigid cohomology, which in contrast to Dwork's theory is closely related to classical (singular) cohomology and applies to arbitrary varieties. We thereby clarify the relation of Lauder's work to Griffiths' theory of periods and to Kedlaya's approach on point counting via Monsky-Washnitzer cohomology. In the case of curves, this also removes Lauder's restriction that the degree has to be prime to the characteristic.