Distinguished-root formulas for generalized Calabi–Yau hypersurfaces

Adolphson, Alan; Sperber, Steven

doi:10.2140/ant.2017.11.1317

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Distinguished-root formulas for generalized Calabi–Yau hypersurfaces

Alan Adolphson and Steven Sperber

Vol. 11 (2017), No. 6, 1317–1356

DOI: 10.2140/ant.2017.11.1317

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 $ℱ$ . That function $ℱ$ is the $p$ -adic analytic continuation of the ratio $F (Λ) ∕ F (Λ^{p})$ , where $F (Λ)$ is a solution of the $A$ -hypergeometric system of differential equations corresponding to the Picard–Fuchs equation of the family.

Keywords

zeta function, Calabi–Yau, $A$-hypergeometric system, $p$-adic analytic function

Mathematical Subject Classification 2010

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

Vol. 11, No. 6, 2017