Distinguished-root formulas for generalized Calabi-Yau hypersurfaces

By a “generalized Calabi-Yau hypersurface” we mean a hypersurface in ℙⁿ of degree d dividing n C1. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal p-divisibility. We study the p-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of p times a product of special values of a certain p-adic analytic function F. That function F is the p-adic analytic continuation of the ratio F.(Λ)=F.(Λ^p), where F.(Λ) is a solution of the A-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.