Frobenius Distributions of Low Dimensional Abelian Varieties Over Finite Fields

Abstract

Given a g-dimensional abelian variety A over a finite field Fq, 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 ≤ 3. We also give a partial classification for simple ordinary abelian varieties of prime dimension g≥3.