Honda-Tate Theorem (reviewed)

Let $A$ and $B$ be abelian varieties over $\mathbb{F}_q$. Let $P_A(t) = \det(t-F|H^1(A))$ and $P_B(t) = \det(t-F|H^1(B))$ be the characteristic polynomial of the action of the Frobenius endomorphism on the first cohomology group of $A$ and $B$ respectively. The following are equivalent:

1) $P_A(t)$ divides $P_B(t)$: $P_A(t)\vert P_B(t)$.

2) A is an isogeny factor of $B$: $A \sim B' \leq B$.

Moreover, one can characterize the polynomials that can occur as $P_A(t)$ for some $A$: they must be Weil polynomials and there is a further condition on the multiplicities of factors described in: Waterhouse, Abelian varieties over finite fields [MR:265369, 10.24033/asens.1183].