Elliptic K3 surfaces associated with the product of two elliptic curves: Mordell-Weil lattices and their fields of definition

TL;DR

This paper explicitly describes the Mordell-Weil groups of elliptic K3 surfaces associated with pairs of elliptic curves, using geometric and group-theoretic methods, and computes these groups in several arithmetic examples.

Contribution

It provides explicit descriptions and computational methods for Mordell-Weil groups of elliptic K3 surfaces linked to pairs of elliptic curves, including non-generic cases.

Findings

Explicit Mordell-Weil group descriptions for generic cases.

A method to compute finite index subgroups in non-generic cases.

Full Mordell-Weil groups computed for examples with complex multiplication.

Abstract

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Kuwata. We give an explicit description of the geometric Mordell-Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the non-generic case, we describe a method to calculate explicitly a finite index subgroup of the Mordell-Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell-Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.