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The effect of twisting on the 2-Selmer group


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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[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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