Distinguished-root formulas for generalized Calabi–Yau hypersurfaces

Abstract

By a “generalized Calabi–Yau hypersurface” we mean a hypersurface in ${\mathbb{P}}^n$ of degree $d$ dividing $n+1$. 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 $\mathcal{ℱ}$. That function $\mathcal{ℱ}$ is the $p$-adic analytic continuation of the ratio $F\left(\Lambda \right)∕F\left({\Lambda }^p\right)$, where $F\left(\Lambda \right)$ is a solution of the $A$-hypergeometric system of differential equations corresponding to the Picard–Fuchs equation of the family.