Cyclicity of CM elliptic curves modulo 𝑝

Abstract

Let E E be an elliptic curve defined over Q \mathbb {Q} and with complex multiplication. For a prime p p of good reduction, let E ¯ \overline {E} be the reduction of E E modulo p . p. We find the density of the primes p ≤ x p \leq x for which E ¯ ( F p ) \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.