Papers › Frobenius distributions of low dimensional abelian varieties over finite fields
Frobenius distributions of low dimensional abelian varieties over finite fields
Santiago Arango-Piñeros, Deewang Bhamidipati, Soumya Sankar
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Given a g-dimensional abelian variety A over a finite field 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 ≤3. We also give a partial classification for simple ordinary abelian varieties of prime dimension g>3.
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