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The effect of twisting on the 2-Selmer group

  • PETER SWINNERTON–DYER (a1)
Abstract
Abstract

Let Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb be its quadratic twist by b. Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γb as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.

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[1]Colliot–Thélène J.-L., Skorobogatov A. N. and Swinnerton–Dyer Sir P. Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Invent. Math. 134 (1998), 579650.
[2]Heath–Brown R. The size of Selmer groups for the congruent number problem, II. Invent. Math. 118 (1994), 331370.
[3]Kramer K. Arithmetic of elliptic curves upon quadratic extension. Trans. Amer. Math. Soc. 264 (1981), 121135.
[4]Norris J. R. Markov Chains (Cambridge, 1997).
[5]Skorobogatov A. N. and Swinnerton–Dyer Sir P. 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198 (2005), 448483.
[6]Swinnerton–Dyer Sir P. 2-descent through the ages. in Ranks of Elliptic Curves and Random Matrix Theory (ed. Conrey J. B. et al. .), London Math. Soc. Lecture Note Ser. 341 (Cambridge University Press, 2007), 345356.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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