Cyclicity of CM elliptic curves modulo 𝑝

Cojocaru, Alina

doi:10.1090/S0002-9947-03-03283-5

Cyclicity of CM elliptic curves modulo $p$
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by Alina Carmen Cojocaru PDF

Trans. Amer. Math. Soc. 355 (2003), 2651-2662 Request permission

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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