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

Philip Candelas, Xenia de la Ossa
2007 Communications in Number Theory and Physics  
In this article we review the observation, due originally to Dwork, that the ζ-function of a variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of the family of quintic 3-folds, x i = 0, and study the ζ-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
more » ... on certain cohomology groups. The ϕ-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 variety. The periods are defined by series that converge for ϕ p < 1. The values of ϕ that are of interest are those for which ϕ p = ϕ so, for nonzero ϕ, we have ϕ 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.
doi:10.4310/cntp.2007.v1.n3.a2 fatcat:dpk3vjuxqnegfas23delwdv47i