PlanetMath: Mazur's theorem on torsion of elliptic curves

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Mazur's theorem on torsion of elliptic curves

(Theorem)

Theorem 1 (Mazur) Let $E/\mathbb{Q}$ be an elliptic curve. Then the torsion subgroup $E_{\operatorname{torsion}}(\mathbb{Q})$ is exactly one of the following groups:

$\displaystyle \mathbb{Z}/N\mathbb{Z}\quad 1\leq N \leq 10\quad or\quad N=12$

$\displaystyle \mathbb{Z}/2 \mathbb{Z}\oplus \mathbb{Z}/ 2N \mathbb{Z}\quad 1\leq N\leq 4$

Note: see Nagell-Lutz theorem for an efficient algorithm to compute the torsion subgroup of an elliptic curve defined over $\mathbb{Q}$ .

Bibliography

1: Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
2: Barry Mazur, Modular curves and the Eisenstein ideal, IHES Publ. Math. 47 (1977), 33-186.
3: Barry Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.

"Mazur's theorem on torsion of elliptic curves" is owned by alozano.

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Also defines:

Mazur's theorem

Keywords:

torsion, elliptic curve

Attachments:: Nagell-Lutz theorem (Theorem) by alozano
examples of torsion subgroups of elliptic curves (Example) by alozano

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Cross-references: algorithm, Nagell-Lutz theorem, groups, torsion subgroup, elliptic curve
There are 3 references to this entry.

This is version 2 of Mazur's theorem on torsion of elliptic curves, born on 2003-08-18, modified 2003-08-18.
Object id is 4607, canonical name is MazursTheoremOnTorsionOfEllipticCurves.
Accessed 4303 times total.

Classification:

AMS MSC:

14H52 (Algebraic geometry :: Curves :: Elliptic curves)

Pending Errata and Addenda

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