### Abstract

Consider A an abelian variety of dimension r defined over ℚ. Assume that rank_{ℚ} A ≥ g, where g ≥ 0 is an integer, and let a_{1},⋯, a_{g} ϵ A(ℚ) be linearly independent points. (So, in particular, a_{1},⋯, a_{g} have infinite order, and if g = 0, then the set {a_{1},⋯, a_{g}} is empty.) For p a rational prime of good reduction for A, let Ā be the reduction of A at p, let ā_{i} for i=1,⋯, g be the reduction of a_{i} (modulo p), and let 〈ā_{1},⋯, ā_{g}〉 be the subgroup of Ā(F_{p}) generated by ā_{1},⋯, ā_{g}. Assume that ℚ(A[2]) = ℚ and Q(A[2], 2^{-1} a_{1},⋯, 2^{-1}a_{g}) ≠ ℚ. (Note that this particular assumption ℚ(A[2]) = ℚ forces the inequality g ≥ 1, but we can take care of the case g =0, under the right assumptions, also.) Then in this article, in particular, we show that the number of primes p for which Ā(F_{p})/〈ā1,⋯,ā_{g}〉 has at most (2r - 1) cyclic components is infinite. This result is the right generalization of the classical Artin's primitive root conjecture in the context of general abelian varieties; that is, this result is an unconditional proof of Artin's conjecture for abelian varieties. Artin's primitive root conjecture (1927) states that, for any integer a ≠ ±1 or a perfect square, there are infinitely many primes p for which a is a primitive root (mod p). (This conjecture is not known for any specific a.).

Original language | English |
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Pages (from-to) | 737-743 |

Number of pages | 7 |

Journal | Kyoto Journal of Mathematics |

Volume | 56 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 Dec |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyoto Journal of Mathematics*,

*56*(4), 737-743. https://doi.org/10.1215/21562261-3664896