1
$\begingroup$

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...

$\endgroup$
2
  • 2
    $\begingroup$ What do you think are the definitions of these two endomorphisms? $\endgroup$
    – Will Sawin
    Jul 17, 2017 at 15:27
  • 1
    $\begingroup$ I found the introduction of Deligne's Weil I very helpful for this kind of thing. $\endgroup$ Jul 17, 2017 at 15:28

1 Answer 1

2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.