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Title: Zeta functions on projective hypersurfaces via controlled reduction
Abstract: The zeta function of a variety in characteristic p captures a lot of arithmetic information about that variety. Calculating this zeta function as fast as possible is a classical problem in computational number theory. We describe a cohomological approach to this problem, 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, including an "abstract controlled reduction problem" which abstracts the algorithm away from the specifics of any particular class of varieties. 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.