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 a(1), ..., a(g) is an element of A(Q) be linearly independent points. (So, in particular, a(1), ..., a(h) have infinite order, and if g = 0, then the set {a(1), ..., a(g)} is empty.) For p a rational prime of good reduction for A, let (A) over bar be the reduction of A at p, let (a) over bar (i) for i = 1, ..., g be the reduction of a(i) (modulo p), and let <(a) over bar (1), ..., (a) over bar (g)> be the subgroup of (A) over bar (F-p) generated by (a) over bar (1), ..., (a) over bar (g). Assume that Q(A[2]) = Q and Q(A[2],2(-1) a(1,) ..., 2(-1) a(g)) not equal 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 assiimptions, also.) Then in this article, in particular, we show that the number of primes p for which (A) over bar (FP)/<(a) over bar (1), ..., (a) over bar (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 not equal +/- 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.)
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