Mazur's torsion theorem

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In algebraic geometry, Mazur's torsion theorem, due to Barry Mazur, classifies the possible torsion subgroups of the group of rational points on an elliptic curve defined over the rational numbers.

If Cn denotes the cyclic group of order n, then the possible torsion subgroups are Cn with 1 ≤ n ≤ 10, and also C12; and the direct sum of C2 with C2, C4, C6 or C8.

In the opposite direction, all these torsion structures occur infinitely often over Q, since the corresponding modular curves are all genus zero curves with a rational point.

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