Communications in Number Theory and Physics

Volume 1 (2007)

Number 3

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

Pages: 479 – 512



Philip Candelas (Mathematical Institute, Oxford University, United Kingdom)

Xenia de la Ossa (Mathematical Institute, Oxford University, United Kingdom)


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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