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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Torsion on elliptic curves in isogeny classes


Authors: Yasutsugu Fujita and Tetsuo Nakamura
Journal: Trans. Amer. Math. Soc. 359 (2007), 5505-5515
MSC (2000): Primary 11G05
Published electronically: May 11, 2007
MathSciNet review: 2327039
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Abstract: Let $ E$ be an elliptic curve over a number field $ K$ and $ \mathcal C$ its $ K$-isogeny class. We are interested in determining the orders and the types of torsion groups $ E(K)_{\textrm{tors}}$ in $ \mathcal C$. For a prime $ l$, we give the range of possible types of $ l$-primary parts $ E(K)_{(l)}$ of $ E(K)_{\textrm{tors}}$ when $ E$ runs over $ \mathcal C$. One of our results immediately gives a simple proof of a theorem of Katz on the order $ \sup_{E \in \mathcal C}\vert E(K)_{(l)}\vert$ of maximal $ l$-primary torsion in $ \mathcal C$.


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Additional Information

Yasutsugu Fujita
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Email: fyasut@yahoo.co.jp

Tetsuo Nakamura
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
Email: nakamura@math.tohoku.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04212-2
PII: S 0002-9947(07)04212-2
Keywords: Elliptic curve, torsion, isogeny
Received by editor(s): February 27, 2004
Received by editor(s) in revised form: October 24, 2005
Published electronically: May 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.