Kyoto Journal of Mathematics

Artin’s conjecture for abelian varieties

Cristian Virdol

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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.)

Article information

Kyoto J. Math., Volume 56, Number 4 (2016), 737-743.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]

abelian varieties Artin’s conjecture primitive-cyclic points


Virdol, Cristian. Artin’s conjecture for abelian varieties. Kyoto J. Math. 56 (2016), no. 4, 737--743. doi:10.1215/21562261-3664896.

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