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Let $A$ be an abelian variety over a finite field $k$. Let $V_\ell(A)$ be its $\ell$-adic Tate module.

We have a natural action of the absolute Galois group of $k$, and thus an action of the Frobenius automorphism of $\overline{k}$ on $V_\ell(A)$.

However, $A$ also has an endomorphism $F:A\to A$ which also acts on $V_\ell(A)$ (only as an endomorphism a priori).

Are these endomorphisms of $V_\ell(A)$ the same? I am sure they are, but why? I have seen this used many times, and I'm slightly confused about it...

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    $\begingroup$ What do you think are the definitions of these two endomorphisms? $\endgroup$
    – Will Sawin
    Commented Jul 17, 2017 at 15:27
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    $\begingroup$ I found the introduction of Deligne's Weil I very helpful for this kind of thing. $\endgroup$
    – Martin Bright
    Commented Jul 17, 2017 at 15:28

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Does http://www.mathematik.uni-regensburg.de/Jannsen/home/Weil-gesamt-eng.pdf p. 29, Theorem 5.4 help you? One has $T_\ell A = H^1_\mathrm{et}(\bar{A},\mathbf{Z}_\ell)^\vee$.

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