Consider E a CM elliptic curve over Q. Assume that rankQ E ≥ 1, and let a ∈ E(Q) be a point of infinite order. For p a rational prime, we denote by Fp the residue field at p. If E has good reduction at p, let Ē be the reduction of E at p, let ā be the reduction of a(modulo p), and let <ā> be the subgroup of Ē(Fp) generated by ā. Assume that Q(E) = Q and Q(E, 2−1a) = Q. Then in this article we obtain an asymptotic formula for the number of rational primes p, with p ≤ x, for which Ē(Fp)/<ā> is cyclic, and we prove that the number of primes p, for which Ē(Fp)/<ā> is cyclic, is infinite. This result is a generalization of the classical Artin's primitive root conjecture, in the context of CM elliptic curves; that is, this result is an unconditional proof of Artin's primitive root conjecture for CM elliptic curves. Artin's conjecture states that, for any integer a = ±1 or a perfect square (or equivalently a = ±1, and Q(±1, √a) = Q(1, 2−1a) = Q), there are infinitely many primes p for which a is a primitive root (mod p), and an asymptotic formula for such primes is satisfied (this conjecture is not known for any specific a).
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Acknowledgments. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by Ministry of Education grant 2015R1D1A1A01056643.
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