On Artin’s conjecture for CM elliptic curves

Abstract

Consider $E$ a CM elliptic curve over $\mathbb{Q}$. Assume that ${rank}_{\mathbb{Q}}E\ge 1$, and let $a\in E\left(\mathbb{Q}\right)$ be a point of infinite order. For $p$ a rational prime, we denote by ${\mathbb{F}}_p$ the residue field at $p$. If $E$ has good reduction at $p$, let $\overline{E}$ be the reduction of $E$ at $p$, let $\overline{a}$ be the reduction of $a(modulop$), and let $\langle \overline{a}\rangle$ be the subgroup of $\overline{E}\left({\mathbb{F}}_p\right)$ generated by $\overline{a}$. Assume that $\mathbb{Q}\left(E\right[2\left]\right)=\mathbb{Q}$ and $\mathbb{Q}\left(E\right[2],2^{-1}a)\ne \mathbb{Q}$. Then in this article we obtain an asymptotic formula for the number of rational primes $p$, with $p\le x$, for which $\overline{E}\left({\mathbb{F}}_p\right)/\langle \overline{a}\rangle$ is cyclic, and we prove that the number of primes $p$, for which $\overline{E}\left({\mathbb{F}}_p\right)/\langle \overline{a}\rangle$ 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 $a\ne \pm 1$ or a perfect square (or equivalently $a\ne \pm 1$, and $\mathbb{Q}(\pm 1,\sqrt{a})=\mathbb{Q}\left(1\right[2],2^{-1}a)\ne \mathbb{Q}$), 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 $a$).