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)

 
 

 

Cyclicity of CM elliptic curves modulo $p$


Author: Alina Carmen Cojocaru
Journal: Trans. Amer. Math. Soc. 355 (2003), 2651-2662
MSC (2000): Primary 11G05; Secondary 11N36, 11G15, 11R45
DOI: https://doi.org/10.1090/S0002-9947-03-03283-5
Published electronically: March 14, 2003
MathSciNet review: 1975393
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ E $ be an elliptic curve defined over $\mathbb{Q} $ and with complex multiplication. For a prime $ p $ of good reduction, let $\overline{E} $ be the reduction of $ E $ modulo $ p. $ We find the density of the primes $ p \leq x $ for which $ \overline{E}(\mathbb{F} _p) $ is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.


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

  • [acC1] A. C. Cojocaru, ``On the cyclicity of the group of $\mathbb{F} _p$-rational points of non-CM elliptic curves", Journal of Number Theory, vol. 96, no. 2, October 2002, pp. 335-350.
  • [acC2] A. C. Cojocaru, ``Cyclicity of elliptic curves modulo $p$", Ph.D. thesis, Queen's University, Kingston, Canada, 2002.
  • [BMP] I. Borosh, C. J. Moreno, and H. Porta, ``Elliptic curves over finite fields II", Mathematics of Computation, vol. 29, July 1975, pp. 951-964. MR 53:8067
  • [Ho] C. Hooley, ``Applications of sieve methods to the theory of numbers", Cambridge University Press, 1976. MR 53:7976
  • [LT1] S. Lang and H. Trotter, ``Frobenius distributions in $\operatorname{GL}_2$-extensions", Lecture Notes in Mathematics 504, Springer-Verlag, 1976. MR 58:27900
  • [LT2] S. Lang and H. Trotter, ``Primitive points on elliptic curves", Bulletin of the American Mathematical Society, vol. 83, no. 2, March 1977, pp. 289-292. MR 55:308
  • [Mu1] M. Ram Murty, ``On Artin's conjecture", Journal of Number Theory, vol. 16, no. 2, April 1983, pp. 147-168. MR 86f:11087
  • [Mu2] M. Ram Murty, ``An analogue of Artin's conjecture for abelian extensions'', Journal of Number Theory, vol. 18, no. 3, June 1984, pp. 241-248. MR 85j:11161
  • [Mu3] M. Ram Murty, ``Artin's conjecture and elliptic analogues", Sieve Methods, Exponential Sums and their Applications in Number Theory (eds. G. R. H. Greaves, G. Harman, M. N. Huxley), Cambridge University Press, 1996, pp. 326-344. MR 2000a:11098
  • [Mu4] M. Ram Murty, ``Problems in analytic number theory", Graduate Texts in Mathematics 206, Springer-Verlag, 2001. MR 2001k:11002
  • [Sch] W. Schaal, ``On the large sieve method in algebraic number fields", Journal of Number Theory 2, 1970, pp. 249-270. MR 42:7626
  • [Se1] J. -P. Serre, ``Résumé des cours de 1977-1978", Annuaire du Collège de France 1978, pp. 67-70.
  • [Se2] J. -P. Serre, ``Quelques applications du théorème de densité de Chebotarev", Inst. Hautes Etudes Sci. Publ. Math., no. 54, 1981, pp. 123-201. MR 83k:12011
  • [Silv1] J. H. Silverman, ``The arithmetic of elliptic curves", Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986. MR 87g:11070
  • [Silv2] J. H. Silverman, ``Advanced topics in the arithmetic of elliptic curves", Graduate Texts in Mathematics 151, Springer-Verlag, New York, 1994. MR 96b:11074

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G05, 11N36, 11G15, 11R45

Retrieve articles in all journals with MSC (2000): 11G05, 11N36, 11G15, 11R45


Additional Information

Alina Carmen Cojocaru
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6
Address at time of publication: The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario, M5T 3J1, Canada
Email: alina@mast.queensu.ca, alina@fields.utoronto.ca

DOI: https://doi.org/10.1090/S0002-9947-03-03283-5
Keywords: Cyclicity of elliptic curves modulo $p$, complex multiplication, applications of sieve methods
Received by editor(s): July 24, 2002
Received by editor(s) in revised form: December 4, 2002
Published electronically: March 14, 2003
Additional Notes: Research partially supported by an Ontario Graduate Scholarship
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society