Cyclic torsion of elliptic curves

Nakamura, Tetsuo

doi:10.1090/S0002-9939-99-04689-4

Cyclic torsion of elliptic curves
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Proc. Amer. Math. Soc. 127 (1999), 1589-1595 Request permission

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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