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# 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

#### 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.

Received 16 September 2016

Accepted 18 August 2017

Published 27 April 2018