## 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 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(F_{p}) generated by ā. 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 Ē(F_{p})/<ā> is cyclic, and we prove that the number of primes p, for which Ē(F_{p})/<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−^{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).

Original language | English |
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Pages (from-to) | 1361-1371 |

Number of pages | 11 |

Journal | Kyoto Journal of Mathematics |

Volume | 60 |

Issue number | 4 |

DOIs | |

Publication status | 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.

Publisher Copyright:

© 2020 by Kyoto University

## All Science Journal Classification (ASJC) codes

- Mathematics(all)