Artin's conjecture for abelian varieties

Cristian Virdol

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Abstract

Consider A an abelian variety of dimension r defined over ℚ. Assume that rank A ≥ g, where g ≥ 0 is an integer, and let a1,⋯, ag ϵ 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 Ā be the reduction of A at p, let āi for i=1,⋯, g be the reduction of ai (modulo p), and let 〈ā1,⋯, āg〉 be the subgroup of Ā(Fp) generated by ā1,⋯, āg. Assume that ℚ(A[2]) = ℚ and Q(A[2], 2-1 a1,⋯, 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 Ā(Fp)/〈ā1,⋯,āg〉 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.).

Original languageEnglish
Pages (from-to)737-743
Number of pages7
JournalKyoto Journal of Mathematics
Volume56
Issue number4
DOIs
Publication statusPublished - 2016 Dec

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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