Zeta functions on projective hypersurfaces via controlled reduction

3:00pm, Wednesday 13 May 2026

Abstract

We describe a cohomological approach to this problem of computing the zeta function of a variety in characteristic p, which involves a p-adic formula for the Frobenius action on cohomology. Costa and Harvey came up with a fast algorithm called *controlled reduction* which uses this method and is the state of the art for varieties of dimension greater than one. We describe various ways one can improve the algorithms of Costa and Harvey. Using our algorithms, we find many examples of varieties with interesting arithmetic invariants (like Newton polygons and domino numbers). All work is joint with Batubara, Huang, and Mellberg.