## 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.