Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Minimal torsion in isogeny classes of elliptic curves


Author: Raymond Ross
Journal: Trans. Amer. Math. Soc. 344 (1994), 203-215
MSC: Primary 11G05; Secondary 11G07
DOI: https://doi.org/10.1090/S0002-9947-1994-1250824-8
MathSciNet review: 1250824
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let K be a field finitely generated over its prime field, and let $w(K)$ denote the number of roots of unity in K. If K is of characteristic 0, then there is an integer D, divisible only by those primes dividing $w(K)$, such that for any elliptic curve $E/K$ without complex multiplication over K, there is an elliptic curve $E\prime /K$ isogenous to E such that $E\prime {(K)_{{\text {tors}}}}$ is of order dividing D. In case K admits a real embedding, we show $D = 2$, and a nonuniform result is proved in positive characteristic.


References [Enhancements On Off] (What's this?)

  • B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, 1975. MR 0376533
  • S. Kamienny, Torsion points on elliptic curves and $q$-coefficients of modular forms, Invent. Math. 109 (1992), no. 2, 221–229. MR 1172689, DOI https://doi.org/10.1007/BF01232025
  • S. Kamienny and B. Mazur, Rational torsion of prime order in elliptic curves over number fields, AstĂ©risque 228 (1995), 3, 81–100. With an appendix by A. Granville; Columbia University Number Theory Seminar (New York, 1992). MR 1330929
  • Nicholas M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481–502. MR 604840, DOI https://doi.org/10.1007/BF01394256
  • Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
  • Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
  • Ju. I. Manin, The $p$-torsion of elliptic curves is uniformly bounded, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465 (Russian). MR 0272786
  • B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI https://doi.org/10.1007/BF01390348
  • AndrĂ© NĂ©ron, Problèmes arithmĂ©tiques et gĂ©omĂ©triques rattachĂ©s Ă  la notion de rang d’une courbe algĂ©brique dans un corps, Bull. Soc. Math. France 80 (1952), 101–166 (French). MR 56951
  • Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
  • Ju. G. Zarhin, Abelian varieties in characteristic $p$, Mat. Zametki 19 (1976), no. 3, 393–400 (Russian). MR 422287

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11G05, 11G07

Retrieve articles in all journals with MSC: 11G05, 11G07


Additional Information

Article copyright: © Copyright 1994 American Mathematical Society