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 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 . That function is the p-adic analytic continuation of the ratio F(Λ)F(Λp), where F(Λ) 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
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