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December 2016 Artin’s conjecture for abelian varieties
Cristian Virdol
Kyoto J. Math. 56(4): 737-743 (December 2016). DOI: 10.1215/21562261-3664896


Consider A an abelian variety of dimension r defined over Q. Assume that rankQAg, where g0 is an integer, and let a1,,agA(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],21a1,,21ag)Q. (Note that this particular assumption Q(A[2])=Q forces the inequality g1, 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 (2r1) 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.)


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Cristian Virdol. "Artin’s conjecture for abelian varieties." Kyoto J. Math. 56 (4) 737 - 743, December 2016.


Received: 25 December 2014; Revised: 17 September 2015; Accepted: 24 September 2015; Published: December 2016
First available in Project Euclid: 7 November 2016

zbMATH: 1379.11064
MathSciNet: MR3568639
Digital Object Identifier: 10.1215/21562261-3664896

Primary: 11G10 , 11G15

Keywords: abelian varieties , Artin’s conjecture , primitive-cyclic points

Rights: Copyright © 2016 Kyoto University


Vol.56 • No. 4 • December 2016
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