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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 145, Issue 3
  • November 2008, pp. 513-526

The effect of twisting on the 2-Selmer group

  • DOI:
  • Published online: 01 November 2008

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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[2]R. Heath–Brown The size of Selmer groups for the congruent number problem, II. Invent. Math. 118 (1994), 331370.

[3]K. Kramer Arithmetic of elliptic curves upon quadratic extension. Trans. Amer. Math. Soc. 264 (1981), 121135.

[5]A. N. Skorobogatov and Sir P. Swinnerton–Dyer 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198 (2005), 448483.

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Mathematical Proceedings of the Cambridge Philosophical Society
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  • EISSN: 1469-8064
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