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 $F(\Lambda )/F({\Lambda }^p)$, where $F(\Lambda )$ is a solution of the $A$-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.

33 pages

Cited by in corpus (2)

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