Mazur’s theorem on torsion of elliptic curves

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

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

\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 $\mathbb{Q}$ .

References

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.

Title	Mazur’s theorem on torsion of elliptic curves
Canonical name	MazursTheoremOnTorsionOfEllipticCurves
Date of creation	2013-03-22 13:51:59
Last modified on	2013-03-22 13:51:59
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Theorem
Classification	msc 14H52
Related topic	EllipticCurve
Related topic	MordellWeilTheorem
Related topic	RankOfAnEllipticCurve
Related topic	TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve
Related topic	ArithmeticOfEllipticCurves
Defines	Mazur’s theorem