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.(Λp), where F.(Λ) is a solution of the A-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.
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- A-hypergeometric system
- P-adic analytic function
- Zeta function