Frobenius distributions of low dimensional abelian varieties over finite fields

Abstract

Given a $g$-dimensional abelian variety $A$ over a finite field $\mathbf{F}_q$, the Weil conjectures imply that the normalized Frobenius eigenvalues generate a multiplicative group of rank at most $g$. The Pontryagin dual of this group is a compact abelian Lie group that controls the distribution of high powers of the Frobenius endomorphism. This group, which we call the Serre--Frobenius group, encodes the possible multiplicative relations between the Frobenius eigenvalues. In this article, we classify all possible Serre--Frobenius groups that occur for $g \le 3$. We also give a partial classification for simple ordinary abelian varieties of prime dimension $g>3$.