The Dwork Family and Hypergeometric Functions

TL;DR

This paper explores the connection between the Dwork family of hypersurfaces and hypergeometric functions, providing computational tools to analyze their differential equations and expand understanding in algebraic geometry and physics.

Contribution

The authors developed a Pari-GP algorithm to verify the hypergeometric nature of differential equations associated with the Dwork family across multiple cases.

Findings

The differential equations are hypergeometric for the Dwork family.

The algorithm successfully computes Gauss-Manin connections.

Supports broader applications in mirror symmetry and deformation theory.

Abstract

In his work studying the Zeta functions of families of hypersurfaces, Dwork came upon a one-parameter family of hypersurfaces (now known as \emph{the} Dwork family). These examples were not only useful to Dwork in his study of his deformation theory for computing Zeta functions of families, but they have also proven to be extremely useful to physicists working in mirror symmetry. A startling result is that these families are very closely linked to hypergeometric functions. This phenomenon was carefully studied by Dwork and Candelas, de la Ossa, and Rodr\'{i}guez-Villegas in a few special cases. Dwork, Candelas, et.al. observed that, for these families, the differential equation associated to the Gauss-Manin connection is in fact hypergeometric. We have developed a computer algorithm, implemented in Pari-GP, which can check this result for more cases by computing the Gauss-Manin connection and the parameters of the hypergeometric differential equation.