## Kyoto Journal of Mathematics

### Artin’s conjecture for abelian varieties

Cristian Virdol

#### Abstract

Consider $A$ an abelian variety of dimension $r$ defined over $\mathbb{Q}$. Assume that $\operatorname{rank}_{\mathbb{Q}}A\geq g$, where $g\geq0$ is an integer, and let $a_{1},\ldots,a_{g}\in A(\mathbb{Q})$ be linearly independent points. (So, in particular, $a_{1},\ldots,a_{g}$ have infinite order, and if $g=0$, then the set $\{a_{1},\ldots,a_{g}\}$ is empty.) For $p$ a rational prime of good reduction for $A$, let $\bar{A}$ be the reduction of $A$ at $p$, let $\bar{a}_{i}$ for $i=1,\ldots,g$ be the reduction of $a_{i}$ (modulo $p$), and let $\langle\bar{a}_{1},\ldots,\bar{a}_{g}\rangle$ be the subgroup of $\bar{A}(\mathbb{F}_{p})$ generated by $\bar{a}_{1},\ldots,\bar{a}_{g}$. Assume that $\mathbb{Q}(A[2])=\mathbb{Q}$ and $\mathbb{Q}(A[2],2^{-1}a_{1},\ldots,2^{-1}a_{g})\neq\mathbb{Q}$. (Note that this particular assumption $\mathbb{Q}(A[2])=\mathbb{Q}$ forces the inequality $g\geq1$, 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 $\frac{\bar{A}(\mathbb{F}_{p})}{\langle\bar{a}_{1},\ldots,\bar{a}_{g}\rangle}$ 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 $a\neq\pm1$ or a perfect square, there are infinitely many primes $p$ for which $a$ is a primitive root $(\operatorname{mod}p)$. (This conjecture is not known for any specific $a$.)

#### Article information

Source
Kyoto J. Math., Volume 56, Number 4 (2016), 737-743.

Dates
Revised: 17 September 2015
Accepted: 24 September 2015
First available in Project Euclid: 7 November 2016

https://projecteuclid.org/euclid.kjm/1478509216

Digital Object Identifier
doi:10.1215/21562261-3664896

Mathematical Reviews number (MathSciNet)
MR3568639

Zentralblatt MATH identifier
1379.11064

#### Citation

Virdol, Cristian. Artin’s conjecture for abelian varieties. Kyoto J. Math. 56 (2016), no. 4, 737--743. doi:10.1215/21562261-3664896. https://projecteuclid.org/euclid.kjm/1478509216

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