Distinguished-root formulas for generalized Calabi-Yau hypersurfaces

TL;DR

This paper investigates the $p$-adic behavior of a special root of the zeta function for generalized Calabi-Yau hypersurfaces, linking it to $p$-adic analytic functions and hypergeometric systems.

Contribution

It establishes a formula for the distinguished root's $p$-adic variation using special values of a $p$-adic analytic function related to hypergeometric solutions.

Findings

The distinguished root's $p$-adic variation equals a product involving $p$ and special values of ${ m extbf{F}}$.

The function ${ m extbf{F}}$ is the $p$-adic continuation of a ratio of hypergeometric solutions.

The results connect zeta function roots with $p$-adic hypergeometric functions.

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.