Artin’s conjecture for abelian varieties

Cristian Virdol (Yonsei)

Abstract

Artin's primitive root conjecture (1927) states that, for any integer \(a \neq \pm 1\) 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 \(a\). In my talk I will prove the equivalent of this conjecture unconditionally for general abelian varieties for all a. Moreover, under GRH, I will prove the strong form of Artin's conjecture (1927) for abelian varieties, i.e. I will prove the density and the asymptotic formula for the primitive primes.