Mazur's torsion theorem

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 C_n denotes the cyclic group of order n, then the possible torsion subgroups are C_n 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 of genus zero.

References

• Mazur, Barry: Rational isogenies of prime degree, Inventiones Math. 44, 2 (June 1978), pp 129–162.