The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists

Candelas, Philip; de la Ossa, Xenia

The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists

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Description

In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a one parameter family of quintic threefolds, and study the zeta-function as a function of the parameter ϕ. Owing to cancellations, the superdeterminant of an infinite matrix reduces to the (ordinary) determinant of a finite matrix, U(ϕ), corresponding to the action of the Frobenius map on certain cohomology groups. The parameter-dependence of U(ϕ) is given by a relation U(ϕ)=E^{-1}(ϕ^p)U(0)E(ϕ) with E(ϕ) a Wronskian matrix formed from the periods of the manifold. The periods are defined by series that converge for $|ϕ|_p < 1$. The values of ϕthat are of interest are those for which ϕ^p = ϕso, for nonzero ϕ, we have |\vph|_p=1. We explain how the process of p-adic analytic continuation applies to this case. The matrix U(ϕ) breaks up into submatrices of rank 4 and rank 2 and we are able from this perspective to explain some of the observations that have been made previously by numerical calculation.
29 pages, 1 figure