Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
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
Published electronically: March 14, 2003
MathSciNet review: 1975393
Full-text PDF Free Access

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?)

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

PII: S 0002-9947(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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia