Categories of abelian varieties over finite fields, I: Abelian varieties over Fp

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