Jacobian elliptic Kummer surfaces and special function identities

TL;DR

This paper explores the structure of Jacobian elliptic fibrations on Kummer surfaces derived from two elliptic curves, linking their period systems to hypergeometric differential equations and establishing new identities among these systems.

Contribution

It introduces a comprehensive method to construct all inequivalent Jacobian elliptic fibrations on Kummer surfaces and derives new identities relating hypergeometric systems.

Findings

Explicit formulas for elliptic fibrations on Kummer surfaces.

Representation of Picard-Fuchs systems as hypergeometric differential equations.

A new identity connecting Appell and Gauss hypergeometric systems.

Abstract

We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities relating differential systems are obtained. They include a new identity relating Appell's hypergeometric system to a product of two Gauss' hypergeometric differential equations by a cubic transformation.