Cyclicity of CM elliptic curves modulo $p$
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- by Alina Carmen Cojocaru PDF
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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.References
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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
- MR Author ID: 703080
- Email: alina@mast.queensu.ca, alina@fields.utoronto.ca
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 1975393