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].

**Knowl status:**

- Review status: reviewed
- Last edited by Edgar Costa on 2020-01-04 09:57:48

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**History:**(expand/hide all)

- 2022-03-26 19:11:33 by Bjorn Poonen
- 2022-03-26 16:01:30 by Bjorn Poonen
- 2020-01-04 09:57:48 by Edgar Costa (Reviewed)
- 2019-05-04 20:31:37 by Kiran S. Kedlaya (Reviewed)
- 2019-05-04 20:24:48 by Kiran S. Kedlaya (Reviewed)
- 2018-05-17 17:21:39 by Kiran S. Kedlaya

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