Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology

Edgar Costa, David Harvey, Kiran S. Kedlay

Published 2018-06-01Version 1

We give an interim report on some improvements and generalizations of the Abbott--Kedlaya--Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi--Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad--Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether--Lefschetz divisor in the sense of Hassett.