  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.
