Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cyclic torsion of elliptic curves

Author: Tetsuo Nakamura
Journal: Proc. Amer. Math. Soc. 127 (1999), 1589-1595
MSC (1991): Primary 11G05.
Published electronically: February 18, 1999
MathSciNet review: 1476380
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Abstract: Let $E$ be an elliptic curve over a number field $k$ such that $\operatorname{End}_{k}E$
$= {\mathbf Z}$ and let $w(k)$ denote the number of roots of unity in $k$. Ross proposed a question: Is $E$ isogenous over $k$ to an elliptic curve $E'/k$ such that $E'(k)_{tors}$ is cyclic of order dividing $w(k)$? A counter-example of this question is given. We show that $E$ is isogenous to $E'/k$ such that $E'(k)_{tors}\,\subset{\mathbf Z}/w(k)^2{\mathbf Z}$. In case $E$ has complex multiplication and $\operatorname{End}_kE={\mathbf Z}$, we obtain certain criteria whether or not $E$ is isogenous to $E'/k$ such that $E'(k)_{tors}\,\subset{\mathbf Z}/2{\mathbf Z}$.

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Additional Information

Tetsuo Nakamura
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Keywords: Elliptic curve, torsion point, isogeny, complex multiplication
Received by editor(s): December 11, 1996
Received by editor(s) in revised form: September 8, 1997
Published electronically: February 18, 1999
Additional Notes: The author was supported by Grant-Aid for Scientific Research No. 09640003, Ministry of Education, Science and Culture, Japan.
Dedicated: Dedicated to Professor Tsuneo Kanno on his seventieth birthday
Communicated by: William W. Adams
Article copyright: © Copyright 1999 American Mathematical Society