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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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[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
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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