3 years ago

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

Edgar Costa, David Harvey, Kiran S. Kedlaya

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.

Publisher URL: http://arxiv.org/abs/1806.00368

DOI: arXiv:1806.00368v3

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