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ON SELMER GROUPS AND TATE–SHAFAREVICH GROUPS FOR ELLIPTIC CURVES y 2 = x 3 − n 3
Author(s) -
Feng Keqin,
Xiong Maosheng
Publication year - 2012
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579312000046
Subject(s) - mathematics , isogeny , elliptic curve , distribution (mathematics) , quadratic equation , twists of curves , group (periodic table) , rank (graph theory) , combinatorics , pure mathematics , schoof's algorithm , mathematical analysis , quarter period , geometry , physics , quantum mechanics
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 E n : y 2 = x 3 − n 3 . 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 is1 2 log log Xas 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.