Elliptic parameters and defining equations for elliptic fibrations on a Kummer surface

By Masato Kuwata, T. Shioda. 2006

59 citations recorded by the source corpus.

We pose the problem to determine explicit defining equations of various elliptic fibrations on a given K3 surface, and study the case of the Kummer surfaces of the product of two elliptic curves. x1. Introduction 1.1. Problem setting Let X be a K3 surface defined over a base field k, and let k(X) denote its function field. Suppose f : X ! P 1 is an elliptic fibration on X with a section O. Then it defines a non-constant function u = f (x) (x 2 X), and hence an element u 2 k(X). We call u the elliptic parameter for the elliptic fibration f . (Actually u is unique only up to the linear fractional transformations, but to fix the idea, we always choose one u. Note that the subfield k(u) of k(X) is uniquely defined by f ). Now let E denote the generic fiber of f . Then E is an elliptic curve defined over k(u) such that the function field k(u)(E) is isomorphic to k(X) as the extensions of k. Problem 1. Given a K3 surface X/k and an elliptic fibration f , determine (i) the elliptic parameter u for f , (ii) the defining equation of the elliptic curve E/k(u), and (iii) the Mordell-Weil lattice (MWL) E(k(u)).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f5e6b6bd18432ef4ad9e09a2d5fb49c52032bf4d

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Citations in the corpus

Lectures on K3 Surfaces (2016, background): “elliptic curves see [342, 457, 464] and [295] for elliptic fibrations of a generic double plane with ramification over six lines.”
K3 surfaces and equations for Hilbert modular surfaces (2012, background): “has a Shioda–Inose structure, making it 2-isogenous with a product of two elliptic curves [Inose 1978; Shioda 2006; Kuwata and Shioda 2008; Clingher and Doran 2007; Elkies ≥ 2015].”
Elliptic Surfaces (2009, background): “In [43] they derive defining equations and elliptic parameters for all fibrations in the last case.”
Elliptic fibrations on a generic Jacobian Kummer surface (2011, background): “Acknowledgements We thank Henry Cohn, Igor Dolgachev, Noam Elkies, Torsten Ekedahl, Steve Kleiman, Shigeyuki Kondo, Barry Mazur, Matthias Schütt and Tetsuji Shioda for many helpful suggestions, and Thomas Rehn and Achill Schürmann for help with their program SymPol.”