Distinguished-root formulas for generalized Calabi-Yau hypersurfaces (Q2399259)

This paper is concerned with the zeta function of a generalised Calabi-Yau hypersurface. A generalized Calabi-Yau hypersurface is 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 the minimal \(p\)-divisibility. This article studies the \(p\)-adic variation of that distinguished root in a family. The main result is that it is equal to the product of an appropriate power of \(p\) times a special value 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 equations of the family.