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

摘要

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