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

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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.

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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:
https://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