Mazur's torsion theorem

In algebraic geometry and number theory, 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 all genus zero curves with a rational point.

References

• Mazur, Barry (1977), "Modular curves and the Eisenstein ideal", Publications Mathématiques de l'IHÉS 47 (1): 33–186, doi:10.1007/BF02684339, MR 0488287 
• Mazur, Barry (1978), with appendix by Dorian Goldfeld, "Rational isogenies of prime degree", Inventiones Mathematicae 44 (2): 129–162, doi:10.1007/BF01390348, MR 0482230