Science in China Series A: Mathematics

, Volume 44, Issue 2, pp 159–167 | Cite as

Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Article

Abstract

Suppose thatE: y 2 =x(x + M) (x + N) is an elliptic curve, whereM N are rational numbers (#0, ±1), and are relatively prime. LetK be a number field of type (2,...,2) with degree 2′. For arbitrary n, the structure of the torsion subgroup E(K) tors of theK-rational points (Mordell group) ofE is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K) tors themselves. The order of E( K)tors is also proved to be a power of 2 for anyn. Besides, for any elliptic curveE over any number field F, it is shown that E( L)tors = E( F) tors holds for almost all extensionsL/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.

Keywords

elliptic curve Mordell group torsion subgroup number field 

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Copyright information

© Science in China Press 2001

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina

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