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arXiv · 1806.00368

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

Edgar Costa·David Harvey·Kiran S. Kedlaya
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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.

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Edgar Costa, David Harvey, Kiran S. Kedlaya. 2018-06-01. Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology. https://doi.org/10.2140/obs.2019.2.221

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