Vol. 2, 2019

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

Edgar Costa, David Harvey and Kiran S. Kedlaya

Vol. 2 (2019), No. 1, 221–238
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 ${\mathbb{F}}_{\phantom{\rule{0.3em}{0ex}}p}$ 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 nonspecial 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.

Keywords
zeta function, toric varieties, Kedlaya's algorithm, $p$-adic methods
Mathematical Subject Classification 2010
Primary: 11G25
Secondary: 11M38, 11Y16, 14G10