#### Vol. 11, No. 6, 2017

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Distinguished-root formulas for generalized Calabi–Yau hypersurfaces

### Alan Adolphson and Steven Sperber

Vol. 11 (2017), No. 6, 1317–1356
##### Abstract

By a “generalized Calabi–Yau hypersurface” we mean a hypersurface in ${ℙ}^{n}$ 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 $\mathsc{ℱ}$. That function $\mathsc{ℱ}$ 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.

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##### Keywords
zeta function, Calabi–Yau, $A$-hypergeometric system, $p$-adic analytic function
Primary: 11G25
Secondary: 14G15
##### Milestones
Received: 30 March 2016
Revised: 20 February 2017
Accepted: 21 April 2017
Published: 16 August 2017
##### Authors
 Alan Adolphson Department of Mathematics Oklahoma State University Stillwater, OK United States Steven Sperber School of Mathematics University of Minnesota Minneapolis, MN United States