Categories of abelian varieties over finite fields I. Abelian varieties over $\mathbb{F}_p$

TL;DR

This paper establishes a categorical equivalence that classifies abelian varieties over finite fields by associating them with certain lattices and Frobenius actions, extending previous classifications to a broader class.

Contribution

It introduces a functorial construction linking abelian varieties over _p ext{F}_p to semisimple Frobenius modules, generalizing existing classification results.

Findings

Provides a categorical equivalence for abelian varieties over _p ext{F}_p

Extends Waterhouse and Deligne's classification to non-ordinary cases

Describes abelian varieties avoiding _p ext{F}_p ext{p} as eigenvalues of Frobenius

Abstract

We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.