Artin's conjecture for abelian varieties

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 a₁,⋯, a_g ϵ A(ℚ) be linearly independent points. (So, in particular, a₁,⋯, a_g have infinite order, and if g = 0, then the set {a₁,⋯, 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 〈ā₁,⋯, ā_g〉 be the subgroup of Ā(F_p) generated by ā₁,⋯, ā_g. Assume that ℚ(A[2]) = ℚ and Q(A[2], 2^-1 a₁,⋯, 2^-1a_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.).