The zeta-function of a $p$-adic manifold, Dwork theory for physicists

In this article we review the observation, due originally to Dwork,that the $\z$-function of a variety, defined originally over thefield with $p$ elements, is a superdeterminant. We review thisobservation in the context of the family of quintic 3-folds,$\sum_{i=1}^5 x_i^5 - \vph \prod_{i=1}^5 x_i\,{=}\,0$, and study the$\z$-function as a function of the parameter $\vph$. Owing tocancellations, the superdeterminant of an infinite matrix reduces tothe (ordinary) determinant of a finite matrix, $U(\vph)$,corresponding to the action of the Frobenius map on certaincohomology groups. The $\vph$-dependence of $U(\vph)$ is given by arelation $U(\vph) = E^{-1}(\vph^p)U(0)E(\vph)$ with $E(\vph)$ aWronskian matrix formed from the periods of the variety. The periodsare defined by series that converge for $\norm{\vph}_p < 1$. Thevalues of $\vph$ that are of interest are those for which $\vph^p =\vph$ so, for nonzero $\vph$, we have $\norm{\vph}_p=1$. We explainhow the process of $p$-adic analytic continuation applies to thiscase. The matrix $U(\vph)$ breaks up into submatrices of rank 4 andrank 2 and we are able from this perspective to explain some of theobservations that have been made previously by numericalcalculation.

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Published 1 January 2007