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

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.

Accepted for publication in Algebra & Number Theory