Artin’s conjecture for abelian varieties

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 $a$.)