Vol. 1, No. 4, 2007

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The zeta function of monomial deformations of Fermat hypersurfaces

Remke Kloosterman

Vol. 1 (2007), No. 4, 421–450
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