2-Descent on elliptic curves and rational points on certain Kummer surfaces |
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Authors: | Alexei Skorobogatov |
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Affiliation: | a Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK b Department of Pure Mathematics & Statistics, University of Cambridge, Cambridge CB2 ITQ, UK |
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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. |
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Keywords: | Kummer surfaces 2-descent |
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