Cyclicity of CM elliptic curves modulo
Author:
Alina Carmen Cojocaru
Journal:
Trans. Amer. Math. Soc. 355 (2003), 26512662
MSC (2000):
Primary 11G05; Secondary 11N36, 11G15, 11R45
Published electronically:
March 14, 2003
MathSciNet review:
1975393
Fulltext 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 nonCM elliptic curves", Journal of Number Theory, vol. 96, no. 2, October 2002, pp. 335350.
 [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
(53 #8067), http://dx.doi.org/10.1090/S00255718197504042643
 [Ho]
C.
Hooley, Applications of sieve methods to the theory of
numbers, Cambridge University Press, CambridgeNew YorkMelbourne,
1976. Cambridge Tracts in Mathematics, No. 70. MR 0404173
(53 #7976)
 [LT1]
Serge
Lang and Hale
Trotter, Frobenius distributions in
𝐺𝐿₂extensions, Lecture Notes in Mathematics,
Vol. 504, SpringerVerlag, BerlinNew York, 1976. Distribution of Frobenius
automorphisms in 𝐺𝐿₂extensions of the rational
numbers. MR
0568299 (58 #27900)
 [LT2]
S.
Lang and H.
Trotter, Primitive points on elliptic
curves, Bull. Amer. Math. Soc.
83 (1977), no. 2,
289–292. MR 0427273
(55 #308), http://dx.doi.org/10.1090/S000299041977143103
 [Mu1]
M.
Ram Murty, On Artin’s conjecture, J. Number Theory
16 (1983), no. 2, 147–168. MR 698163
(86f:11087), http://dx.doi.org/10.1016/0022314X(83)900392
 [Mu2]
M.
Ram Murty, An analogue of Artin’s conjecture for abelian
extensions, J. Number Theory 18 (1984), no. 3,
241–248. MR
746861 (85j:11161), http://dx.doi.org/10.1016/0022314X(84)900593
 [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
(2000a:11098), http://dx.doi.org/10.1017/CBO9780511526091.022
 [Mu4]
M.
Ram Murty, Problems in analytic number theory, Graduate Texts
in Mathematics, vol. 206, SpringerVerlag, New York, 2001. Readings in
Mathematics. MR
1803093 (2001k:11002)
 [Sch]
Werner
Schaal, On the large sieve method in algebraic number fields,
J. Number Theory 2 (1970), 249–270. MR 0272745
(42 #7626)
 [Se1]
J. P. Serre, ``Résumé des cours de 19771978", Annuaire du Collège de France 1978, pp. 6770.
 [Se2]
JeanPierre
Serre, Quelques applications du théorème de
densité de Chebotarev, Inst. Hautes Études Sci. Publ.
Math. 54 (1981), 323–401 (French). MR 644559
(83k:12011)
 [Silv1]
Joseph
H. Silverman, The arithmetic of elliptic curves, Graduate
Texts in Mathematics, vol. 106, SpringerVerlag, New York, 1986. MR 817210
(87g:11070)
 [Silv2]
Joseph
H. Silverman, Advanced topics in the arithmetic of elliptic
curves, Graduate Texts in Mathematics, vol. 151, SpringerVerlag,
New York, 1994. MR 1312368
(96b:11074)
 [acC1]
 A. C. Cojocaru, ``On the cyclicity of the group of rational points of nonCM elliptic curves", Journal of Number Theory, vol. 96, no. 2, October 2002, pp. 335350.
 [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. 951964. 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, SpringerVerlag, 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. 289292. MR 55:308
 [Mu1]
 M. Ram Murty, ``On Artin's conjecture", Journal of Number Theory, vol. 16, no. 2, April 1983, pp. 147168. 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. 241248. 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. 326344. MR 2000a:11098
 [Mu4]
 M. Ram Murty, ``Problems in analytic number theory", Graduate Texts in Mathematics 206, SpringerVerlag, 2001. MR 2001k:11002
 [Sch]
 W. Schaal, ``On the large sieve method in algebraic number fields", Journal of Number Theory 2, 1970, pp. 249270. MR 42:7626
 [Se1]
 J. P. Serre, ``Résumé des cours de 19771978", Annuaire du Collège de France 1978, pp. 6770.
 [Se2]
 J. P. Serre, ``Quelques applications du théorème de densité de Chebotarev", Inst. Hautes Etudes Sci. Publ. Math., no. 54, 1981, pp. 123201. MR 83k:12011
 [Silv1]
 J. H. Silverman, ``The arithmetic of elliptic curves", Graduate Texts in Mathematics 106, SpringerVerlag, New York, 1986. MR 87g:11070
 [Silv2]
 J. H. Silverman, ``Advanced topics in the arithmetic of elliptic curves", Graduate Texts in Mathematics 151, SpringerVerlag, 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:
http://dx.doi.org/10.1090/S0002994703032835
PII:
S 00029947(03)032835
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
