Mazur's theorem on torsion of elliptic curvesMazursTheoremOnTorsionOfEllipticCurves2003-08-18 10:12:082003-08-18 10:37:32TheoremMazur's theoremtorsionelliptic curve% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}\begin{thm}[Mazur]
Let $E/\Rats$ be an elliptic curve. Then the torsion subgroup
$E_{\operatorname{torsion}}(\Rats)$ is exactly one of the
following groups:
$$\Ints/N\Ints \quad 1\leq N \leq 10\quad or\quad N=12$$
$$\Ints /2 \Ints \oplus \Ints / 2N \Ints \quad 1\leq N\leq 4$$
\end{thm}
Note: see Nagell-Lutz theorem for an efficient algorithm to compute the torsion subgroup of an elliptic curve defined over $\Rats$.
\begin{thebibliography}{9}
\bibitem{silverman} Joseph H. Silverman, {\em The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.
\bibitem{mazur1} Barry Mazur, {\em Modular curves and the
Eisenstein ideal}, IHES Publ. Math. 47 (1977), 33-186.
\bibitem{mazur2} Barry Mazur, {\em Rational isogenies of prime
degree}, Invent. Math. 44 (1978), 129-162.
\end{thebibliography}