Consider a CM elliptic curve over . Assume that , and let be a point of infinite order. For a rational prime, we denote by the residue field at . If has good reduction at , let be the reduction of at , let be the reduction of ), and let be the subgroup of generated by . Assume that and . Then in this article we obtain an asymptotic formula for the number of rational primes , with , for which is cyclic, and we prove that the number of primes , for which 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 or a perfect square (or equivalently , and ), there are infinitely many primes for which is a primitive root (mod ), and an asymptotic formula for such primes is satisfied (this conjecture is not known for any specific ).
"On Artin’s conjecture for CM elliptic curves." Kyoto J. Math. 60 (4) 1361 - 1371, December 2020. https://doi.org/10.1215/21562261-2019-0064