# Mazur’s theorem on torsion of elliptic curves

###### 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:

 $\mathbb{Z}/N\mathbb{Z}\quad 1\leq N\leq 10\quad or\quad N=12$
 $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2N\mathbb{Z}\quad 1\leq N\leq 4$

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, . 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