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

Published electronically:
March 14, 2003

MathSciNet review:
1975393

Full-text PDF Free Access

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*, Math. Comput.**29**(1975), 951–964. MR**0404264**, 10.1090/S0025-5718-1975-0404264-3**[Ho]**C. Hooley,*Applications of sieve methods to the theory of numbers*, Cambridge University Press, Cambridge-New York-Melbourne, 1976. Cambridge Tracts in Mathematics, No. 70. MR**0404173****[LT1]**Serge Lang and Hale Trotter,*Frobenius distributions in 𝐺𝐿₂-extensions*, Lecture Notes in Mathematics, Vol. 504, Springer-Verlag, Berlin-New York, 1976. Distribution of Frobenius automorphisms in 𝐺𝐿₂-extensions of the rational numbers. MR**0568299****[LT2]**S. Lang and H. Trotter,*Primitive points on elliptic curves*, Bull. Amer. Math. Soc.**83**(1977), no. 2, 289–292. MR**0427273**, 10.1090/S0002-9904-1977-14310-3**[Mu1]**M. Ram Murty,*On Artin’s conjecture*, J. Number Theory**16**(1983), no. 2, 147–168. MR**698163**, 10.1016/0022-314X(83)90039-2**[Mu2]**M. Ram Murty,*An analogue of Artin’s conjecture for abelian extensions*, J. Number Theory**18**(1984), no. 3, 241–248. MR**746861**, 10.1016/0022-314X(84)90059-3**[Mu3]**M. Ram Murty,*Artin’s conjecture and elliptic analogues*, Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995) London Math. Soc. Lecture Note Ser., vol. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 325–344. MR**1635711**, 10.1017/CBO9780511526091.022**[Mu4]**M. Ram Murty,*Problems in analytic number theory*, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR**1803093****[Sch]**Werner Schaal,*On the large sieve method in algebraic number fields*, J. Number Theory**2**(1970), 249–270. MR**0272745****[Se1]**J. -P. Serre, ``Résumé des cours de 1977-1978", Annuaire du Collège de France 1978, pp. 67-70.**[Se2]**Jean-Pierre Serre,*Quelques applications du théorème de densité de Chebotarev*, Inst. Hautes Études Sci. Publ. Math.**54**(1981), 323–401 (French). MR**644559****[Silv1]**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****[Silv2]**Joseph H. Silverman,*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368**

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:
http://dx.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