Artin’s conjecture for abelian varieties

Abstract

Consider A an abelian variety of dimension r defined over Q. Assume that rankQA≥g, where g≥0 is an integer, and let a1,…,ag∈A(Q) be linearly independent points. (So, in particular, a1,…,ag have infinite order, and if g=0, then the set {a1,…,ag} is empty.) For p a rational prime of good reduction for A, let A¯ be the reduction of A at p, let a¯i for i=1,…,g be the reduction of ai (modulo p), and let 〈a¯1,…,a¯g〉 be the subgroup of A¯(Fp) generated by a¯1,…,a¯g. Assume that Q(A[2])=Q and Q(A[2],2−1a1,…,2−1ag)≠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 A¯(Fp)〈a¯1,…,a¯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 (modp). (This conjecture is not known for any specific a.)