2-Descent on elliptic curves and rational points on certain Kummer surfaces

Abstract

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579–650]. To describe the rest, let E ( 1 ) and E ( 2 ) be elliptic curves, D ( 1 ) and D ( 2 ) their respective 2-coverings, and X be the Kummer surface attached to D ( 1 ) × D ( 2 ) . In the appendix we study the Brauer–Manin obstruction to the existence of rational points on X . In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer–Manin obstructions. This second part depends on the hypothesis that the relevent Tate–Shafarevich group is finite, but it does not require Schinzel's Hypothesis.