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ON SELMER GROUPS AND TATE–SHAFAREVICH GROUPS FOR ELLIPTIC CURVES y2=x3n3

Published online by Cambridge University Press:  12 April 2012

Keqin Feng
Affiliation:
Department of Mathematical Science, Tshinghua University, Beijing 100084, P.R. China (email: kfeng@math.tshinghua.edu.cn)
Maosheng Xiong
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, P.R. China (email: mamsxiong@ust.hk)
Corresponding

Abstract

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers nX is as X. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.

Type
Research Article
Copyright
Copyright © University College London 2012

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ON SELMER GROUPS AND TATE–SHAFAREVICH GROUPS FOR ELLIPTIC CURVES y2=x3n3
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