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

TL;DR

This paper improves the Abbott-Kedlaya-Roe method for efficiently computing zeta functions of nondegenerate hypersurfaces in toric varieties over finite fields, with applications to complex algebraic varieties.

Contribution

It introduces generalizations and enhancements to an existing p-adic cohomology method, enabling linear-time computation of zeta functions for a broader class of hypersurfaces.

Findings

Successfully computed zeta functions for K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold.

Verified that a specific non-special cubic fourfold does not correspond to a Noether-Lefschetz divisor.

Demonstrated the method's efficiency and applicability to complex algebraic geometry examples.

Abstract

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.