Communications in Number Theory and Physics
Volume 12 (2018)
Jacobian elliptic Kummer surfaces and special function identities
Pages: 97 – 125
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.
Received 16 September 2016
Accepted 18 August 2017
Published 27 April 2018