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ON SELMER GROUPS AND TATE–SHAFAREVICH GROUPS FOR ELLIPTIC CURVES y2=x3−n3

Part of: Multiplicative number theory Curves Exponential sums and character sums Arithmetic algebraic geometry

Published online by Cambridge University Press:  12 April 2012

Keqin Feng  and
Maosheng Xiong
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)
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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=x3−n3. 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 n≤X 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.

MSC classification

Secondary: 11G05: Elliptic curves over global fields 14H52: Elliptic curves 11L40: Estimates on character sums 11N45: Asymptotic results on counting functions for algebraic and topological structures
Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012 , pp. 236 - 274
Copyright
Copyright © University College London 2012

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