Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Minimal torsion in isogeny classes of elliptic curves
HTML articles powered by AMS MathViewer

by Raymond Ross PDF
Trans. Amer. Math. Soc. 344 (1994), 203-215 Request permission

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
  • 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 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 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, DOI 10.1515/9781400881710
  • 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 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
  • © Copyright 1994 American Mathematical Society
  • 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