# On Artin's conjecture for CM elliptic curves

Cristian Virdol

Research output: Contribution to journalArticlepeer-review

## Abstract

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[2]) = Q and Q(E[2], 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], 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).

Original language English 1361-1371 11 Kyoto Journal of Mathematics 60 4 https://doi.org/10.1215/21562261-2019-0064 Published - 2020 Dec

### Bibliographical note

Funding Information:
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.