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ON SELMER GROUPS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH A TWO‐TORSION OVER Q
Author(s) -
Xiong Maosheng
Publication year - 2013
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/s0025579312001143
Subject(s) - mathematics , isogeny , twists of curves , elliptic curve , torsion (gastropod) , supersingular elliptic curve , quadratic equation , pure mathematics , distribution (mathematics) , schoof's algorithm , mathematical analysis , quarter period , geometry , medicine , surgery
We study the distribution of the size of Selmer groups arising from a 2‐isogeny and its dual 2‐isogeny for quadratic twists of elliptic curves with a non‐trivial 2‐torsion point over Q . This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553] which studied the same subject for elliptic curves with full 2‐torsions over Q and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curvesy 2 = x 3 ‐ n 3 . Mathematika 58 (2012), 236–274.] for the special elliptic curvesy 2 = x 3 ‐ n 3 . It is shown that the 2‐ranks of these groups all follow the same distribution and in particular, the mean value is1 2 log log Xfor square‐free positive integers n ≤ X as X → ∞ .