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 Fp 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
Milestones
Received: 1 March 2018
Revised: 24 August 2018
Accepted: 4 September 2018
Published: 13 February 2019
Authors
Edgar Costa
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
David Harvey
School of Mathematics and Statistics
University of New South Wales
Sydney
Australia
Kiran S. Kedlaya
Department of Mathematics
University of California, San Diego
La Jolla, CA
United States