Cyclic torsion of elliptic curves
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- by Tetsuo Nakamura PDF
- 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 $E’(k)_{tors} \subset {\mathbf Z}/2{\mathbf Z}$.References
- N. Aoki, Torsion points on abelian varieties with complex multiplication, In Algebraic Cycles and Related Topics, Kitasakado 1994, F. Hazama, ed., World Scientific, Singapole, New Jersey, London, HongKong, 1995, 1-22.
- Raymond Ross, Minimal torsion in isogeny classes of elliptic curves, Trans. Amer. Math. Soc. 344 (1994), no. 1, 203–215. MR 1250824, DOI 10.1090/S0002-9947-1994-1250824-8
- J.-P. Serre, Abelian $l$-adic representations and elliptic curves, Benjamin, New York, 1968.
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
Additional Information
- Tetsuo Nakamura
- Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
- Email: nakamura@math.tohoku.ac.jp
- 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.
- Communicated by: William W. Adams
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1589-1595
- MSC (1991): Primary 11G05
- DOI: https://doi.org/10.1090/S0002-9939-99-04689-4
- MathSciNet review: 1476380
Dedicated: Dedicated to Professor Tsuneo Kanno on his seventieth birthday