Torsion conjecture

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field.

Elliptic curves

The (strong) torsion conjecture has been completely resolved in the case of elliptic curves. Mazur (1977, 1978) proved uniform boundedness for elliptic curves over the rationals. His techniques were generalized by Kamienny (1992) and Kamienny & Mazur (1995), who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, Merel (1996) proved the conjecture for elliptic curves over any number field. The proof centers around a careful study of the rational points on classical modular curves. An effective bound for the size of the torsion group in terms of the degree of the number field was given by Parent (1999).

Mazur provided a complete list of possible torsion subgroups for rational elliptic curves. 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. A complete list of possible torsion groups is also available for elliptic curves over quadratic number fields, and there are substantial partial results for cubic and quartic number fields (Sutherland 2012).

References

• Kamienny, Sheldon (1992). "Torsion points on elliptic curves and $q$-coefficients of modular forms". Inventiones Mathematicae. 109 (2): 221–229. doi:10.1007/BF01232025. MR 1172689.
• Kamienny, Sheldon; Mazur, Barry (1995). With an appendix by A. Granville. "Rational torsion of prime order in elliptic curves over number fields". Astérisque. 228: 81–100. MR 1330929.
• 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
• Merel, Loïc (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]. Inventiones Mathematicae (in French). 124 (1): 437–449. doi:10.1007/s002220050059. MR 1369424.
• Parent, Pierre (1999). "Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres" [Effective bounds for the torsion of elliptic curves over number fields]. Journal für die Reine und Angewandte Mathematik (in French). 506: 85–116. arXiv:alg-geom/9611022. doi:10.1515/crll.1999.009. MR 1665681.
• Sutherland, Andrew V. (2012), Torsion subgroups of elliptic curves over number fields (PDF)