Communications in Number Theory and Physics

Volume 12 (2018)

Number 1

Jacobian elliptic Kummer surfaces and special function identities

Pages: 97 – 125

DOI: http://dx.doi.org/10.4310/CNTP.2018.v12.n1.a4

Authors

Elise Griffin (Department of Mathematics and Statistics, Utah State University, Logan, Utah, U.S.A.)

Andreas Malmendier (Department of Mathematics and Statistics, Utah State University, Logan, Utah, U.S.A.)

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.

Full Text (PDF format)

Received 16 September 2016

Accepted 18 August 2017

Published 27 April 2018