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Cyclicity of CM elliptic curves modulo $p$

Author(s): Alina Carmen Cojocaru
Journal: Trans. Amer. Math. Soc. 355 (2003), 2651-2662.
MSC (2000): Primary 11G05; Secondary 11N36, 11G15, 11R45
Posted: March 14, 2003
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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.

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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: 10.1090/S0002-9947-03-03283-5
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
Posted: March 14, 2003
Additional Notes: Research partially supported by an Ontario Graduate Scholarship
Copyright of article: Copyright 2003, American Mathematical Society

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