# Mazur’s theorem on torsion of elliptic curves

###### Theorem 1 (Mazur).

Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve^{}. Then the torsion subgroup
${E}_{\mathrm{torsion}}\mathit{}\mathrm{(}\mathrm{Q}\mathrm{)}$ is exactly one of the
following groups:

$$\mathbb{Z}/N\mathbb{Z}\mathit{\hspace{1em}}1\le N\le 10\mathit{\hspace{1em}}or\mathit{\hspace{1em}}N=12$$ |

$$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2N\mathbb{Z}\mathit{\hspace{1em}}1\le N\le 4$$ |

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 |