Cyclicity of CM elliptic curves modulo

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

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an elliptic curve defined over and with complex multiplication. For a prime of good reduction, let be the reduction of modulo We find the density of the primes for which 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.

**[acC1]**A. C. Cojocaru, ``On the cyclicity of the group of -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 ", 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 -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**

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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