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
| Field | Number theory |
|---|---|
| Conjectured by | Andrew Ogg |
| Conjectured in | 1973 |
| First proof by | Barry Mazur Sheldon Kamienny Loïc Merel |
| First proof in | 1977–1996 |
The (strong) torsion conjecture first posed by Ogg (1973) has been completely resolved in the case of elliptic curves. Barry 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, Loïc 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 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. A complete list of possible torsion groups is also available for elliptic curves over quadratic number fields. There are substantial partial results for quartic and quintic number fields (Sutherland 2012).
See also
Bombieri–Lang conjecture
References
Kamienny, Sheldon (1992). "Torsion points on elliptic curves and q {\displaystyle q} q-coefficients of modular forms". Inventiones Mathematicae. 109 (2): 221–229. Bibcode:1992InMat.109..221K. doi:10.1007/BF01232025. MR 1172689. S2CID 118750444.
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. S2CID 122609075.
Mazur, Barry (1978), with appendix by Dorian Goldfeld, "Rational isogenies of prime degree", Inventiones Mathematicae, 44 (2): 129–162, Bibcode:1978InMat..44..129M, doi:10.1007/BF01390348, MR 0482230, S2CID 121987166
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. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424. S2CID 3590991.
Ogg, Andrew (1973). "Rational points on certain elliptic modular curves". Proc. Symp. Pure Math. Proceedings of Symposia in Pure Mathematics. 24: 221–231. doi:10.1090/pspum/024/0337974. ISBN 9780821814246.
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). 1999 (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)
Topics in algebraic curves
Rational curves
Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic
Elliptic curves
Analytic theory
Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory
Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications
Elliptic curve cryptography Elliptic curve primality
Higher genus
De Franchis theorem Faltings's theorem Hurwitz's automorphisms theorem Hurwitz surface Hyperelliptic curve
Plane curves
AF+BG theorem Bézout's theorem Bitangent Cayley–Bacharach theorem Conic section Cramer's paradox Cubic plane curve Fermat curve Genus–degree formula Hilbert's sixteenth problem Nagata's conjecture on curves Plücker formula Quartic plane curve Real plane curve
Riemann surfaces
Belyi's theorem Bring's curve Bolza surface Compact Riemann surface Dessin d'enfant Differential of the first kind Klein quartic Riemann's existence theorem Riemann–Roch theorem Teichmüller space Torelli theorem
Constructions
Dual curve Polar curve Smooth completion
Structure of curves
Divisors on curves
Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli
ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms
Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem
Singularities
Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles
Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
