The zeta function of monomial deformations of Fermat hypersurfaces

Kloosterman, Remke

doi:10.2140/ant.2007.1.421

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The zeta function of monomial deformations of Fermat hypersurfaces

Remke Kloosterman

Vol. 1 (2007), No. 4, 421–450

DOI: 10.2140/ant.2007.1.421

Abstract

This paper intends to give a mathematical explanation for results on the zeta function of some families of varieties recently obtained in the context of mirror symmetry. In the process we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in $ℙ^{n}$ .

In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the $p$ -adic Picard–Fuchs equation associated with monomial deformations of Fermat hypersurfaces.

As a byproduct we obtain Poincaré duality for the rigid cohomology of certain singular affine varieties.

Keywords

zeta function, $p$-adic Picard–Fuchs equation, Monsky–Washnitzer cohomology

Mathematical Subject Classification 2000

Primary: 14G10

Secondary: 14G15, 11G25

Milestones

Received: 5 March 2007

Revised: 31 August 2007

Accepted: 8 October 2007

Published: 1 November 2007

Authors

Remke Kloosterman
Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 D-30167 Hannover Germany

Vol. 1, No. 4, 2007