On Artin’s conjecture for CM elliptic curves

Abstract

Consider E a CM elliptic curve over Q . Assume that rank Q E ≥ 1 , and let a ∈ E ( Q ) be a point of infinite order. For p a rational prime, we denote by F p the residue field at p . If E has good reduction at p , let E ¯ be the reduction of E at p , let a ¯ be the reduction of a ( modulo p ), and let 〈 a ¯ 〉 be the subgroup of E ¯ ( F p ) generated by a ¯ . Assume that Q ( E [ 2 ] ) = Q and Q ( E [ 2 ] , 2 − 1 a ) ≠ Q . Then in this article we obtain an asymptotic formula for the number of rational primes p , with p ≤ x , for which E ¯ ( F p ) / 〈 a ¯ 〉 is cyclic, and we prove that the number of primes p , for which E ¯ ( F p ) / 〈 a ¯ 〉 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 ] , 2 − 1 a ) ≠ 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 ).