## Abstract

Consider $A$ an abelian variety of dimension $r$ defined over $\mathbb{Q}$. Assume that ${rank}_{\mathbb{Q}}A\ge g$, where $g\ge 0$ is an integer, and let ${a}_{1},\dots ,{a}_{g}\in A\left(\mathbb{Q}\right)$ be linearly independent points. (So, in particular, ${a}_{1},\dots ,{a}_{g}$ have infinite order, and if $g=0$, then the set $\{{a}_{1},\dots ,{a}_{g}\}$ is empty.) For $p$ a rational prime of good reduction for $A$, let $\overline{A}$ be the reduction of $A$ at $p$, let ${\overline{a}}_{i}$ for $i=1,\dots ,g$ be the reduction of ${a}_{i}$ (modulo $p$), and let $\langle {\overline{a}}_{1},\dots ,{\overline{a}}_{g}\rangle $ be the subgroup of $\overline{A}\left({\mathbb{F}}_{p}\right)$ generated by ${\overline{a}}_{1},\dots ,{\overline{a}}_{g}$. Assume that $\mathbb{Q}\left(A\right[2\left]\right)=\mathbb{Q}$ and $\mathbb{Q}\left(A\right[2],{2}^{-1}{a}_{1},\dots ,{2}^{-1}{a}_{g})\ne \mathbb{Q}$. (Note that this particular assumption $\mathbb{Q}\left(A\right[2\left]\right)=\mathbb{Q}$ forces the inequality $g\ge 1$, 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 $\frac{\overline{A}\left({\mathbb{F}}_{p}\right)}{\langle {\overline{a}}_{1},\dots ,{\overline{a}}_{g}\rangle}$ 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 $a\ne \pm 1$ or a perfect square, there are infinitely many primes $p$ for which $a$ is a primitive root $(modp)$. (This conjecture is not known for any specific $a$.)

## Citation

Cristian Virdol. "Artin’s conjecture for abelian varieties." Kyoto J. Math. 56 (4) 737 - 743, December 2016. https://doi.org/10.1215/21562261-3664896

## Information