# Torsion conjecture

Torsion conjecture (Redirected from Mazur's torsion theorem) Jump to navigation Jump to search For other uniform boundedness conjectures, see Uniform boundedness conjecture.

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points 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. The torsion conjecture has been completely resolved in the case of elliptic curves.

Contents 1 Elliptic curves 2 See also 3 References 4 Bibliography Elliptic curves Torsion conjecture for elliptic curves Field Number theory Conjectured by Beppo Levi Conjectured in 1908 First proof by Barry Mazur Sheldon Kamienny Loïc Merel First proof in 1977–1996 From 1906 to 1911, Beppo Levi published a series of papers investigating the possible finite orders of points on elliptic curves over the rationals.[1] He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups: Cn with 1 ≤ n ≤ 10, where Cn denotes the cyclic group of order n; C12; C2n × C2 with 1 ≤ n ≤ 4, where × denotes the direct sum.

At the 1908 International Mathematical Congress in Rome, Levi conjectured that this is a complete list of torsion groups for elliptic curves over the rationals.[1] The torsion conjecture for elliptic curves over the rationals was independently reformulated by Trygve Nagell (1952) and again by Andrew Ogg (1971), with the conjecture becoming commonly known as Ogg's conjecture.[1] Andrew Ogg (1971) drew the connection between the torsion conjecture for elliptic curves over the rationals and the theory of classical modular curves.[1] In the early 1970s, the work of Gérard Ligozat, Daniel Kubert, Barry Mazur, and John Tate showed that several small values of n do not occur as orders of torsion points on elliptic curves over the rationals.[1] Barry Mazur (1977, 1978) proved the full torsion conjecture 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.[1] An effective bound for the size of the torsion group in terms of the degree of the number field was given by Parent (1999). A complete list of possible torsion groups has also been given 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 Uniform boundedness conjecture for preperiodic points Uniform boundedness conjecture for rational points References ^ Jump up to: a b c d e f Schappacher & Schoof 1996, pp. 64–65. Bibliography Kamienny, Sheldon (1992). "Torsion points on elliptic curves and {displaystyle 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. Nagell, Trygve (1952). "Problems in the theory of exceptional points on plane cubics of genus one". Den 11te Skandinaviske Matematikerkongress, Trondheim 1949, Oslo. Johan Grundt Tanum forlag [no]. pp. 71–76. Ogg, Andrew (1971). "Rational points of finite order on elliptic curves". Inventiones Mathematicae. 22: 105–111. 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. Schappacher, Norbert; Schoof, René (1996), "Beppo Levi and the arithmetic of elliptic curves" (PDF), The Mathematical Intelligencer, 18 (1): 57–69, doi:10.1007/bf03024818, MR 1381581, Zbl 0849.01036 Sutherland, Andrew V. (2012), Torsion subgroups of elliptic curves over number fields (PDF) hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves This number theory-related article is a stub. You can help Wikipedia by expanding it.

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