Mazur's theorem on torsion of elliptic curves

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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 $\mathbb{Q}$ .

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.

Defines:

Mazur's theorem

Keywords:

torsion, elliptic curve

Type of Math Object:

Theorem

Major Section:

Reference