Honda-Tate theorem (reviewed)

The Honda-Tate theorem states that abelian varieties $A$ and $B$ over $\F_q$ are isogenous if and only if they have the same characteristic polynomial, and it characterizes the polynomials that can arise as characteristic polynomials of abelian varieties over $\F_q$ as being the Weil polynomials satisfying a condition on the multiplicities of factors. The condition is given explicitly in Waterhouse, "Abelian varieties over finite fields" [MR:265369, 10.24033/asens.1183].

Moreover, $A$ is isogenous to an abelian subvariety of $B$ if and only $P_A(t)$ divides $P_B(t)$.