Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2007-05-14
Physics
High Energy Physics
High Energy Physics - Theory
29 pages, 1 figure
Scientific paper
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 \phi. Owing to cancellations, the superdeterminant of an infinite matrix reduces to the (ordinary) determinant of a finite matrix, U(\phi), corresponding to the action of the Frobenius map on certain cohomology groups. The parameter-dependence of U(\phi) is given by a relation U(\phi)=E^{-1}(\phi^p)U(0)E(\phi) with E(\phi) a Wronskian matrix formed from the periods of the manifold. The periods are defined by series that converge for $|\phi|_p < 1$. The values of \phi that are of interest are those for which \phi^p = \phi so, for nonzero \phi, we have |\vph|_p=1. We explain how the process of p-adic analytic continuation applies to this case. The matrix U(\phi) 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.
Candelas Philip
la Ossa Xenia de
No associations
LandOfFree
The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-622721