Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = p^n$ and $\ell \neq p$ be a prime.

Is it known if $\dim \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1))^\Gamma = \dim \mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1))^\Gamma$?

Perhaps this follows (for Abelian varieties) from the Tate conjecture in the Hom form $\mathrm{End}(A) \otimes_\mathbf{Z} \mathbf{Q}_\ell = \mathrm{End}_\Gamma(V_\ell A)$ since the left hand side is independent of $\ell$ and this formula holds for $\ell = p$ as well (Zarhin, Homomorphisms of abelian varieties over finite fields)?

Edit: What I know:

  1. $\dim \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1)) = \dim \mathrm{H}^2_{crys}(\bar{X}/W) \otimes \mathbf{Q}_p$ (by Katz-Messing since crystalline cohomology is a Weil cohomology theory)

  2. $\mathrm{H}^2_{et}(X,\mathbf{Q}_\ell(1)) = \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1))^\Gamma$ and $\mathrm{H}^2_{fppf}(X,\mathbf{Q}_p(1)) = \mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1))^\Gamma$ (by the Hochschild-Serre spectral sequence since $\mathrm{cd}(\Gamma) = 1$).

  3. $\dim \mathrm{H}^2_{fppf}(X,\mathbf{Q}_\ell(1)) \geq \rho(X) := \mathrm{rk}(\mathrm{NS}(X))$

  4. $\dim \mathrm{H}^2_{et}(X,\mathbf{Q}_\ell(1)) = \rho(X)$ if $X$ is a product of smooth projective curves and Abelian varieties (by Tate, Endomorphisms of Abelian Varieties over Finite Fields, p. 143, Theorem 4)

  5. $\dim \mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1)) \leq \dim \mathrm{H}^2_{cris}(\bar{X}/W) \otimes \mathbf{Q}_p$ (by Illusie, Complexe de de Rham-Witt et cohomologie crystalline, p. 627, Théorème 5.5 (5.5.3) or p. 631, Théorème 5.14) (one even has an injection $\mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1)) \hookrightarrow \mathrm{H}^2_{cris}(\bar{X}/W) \otimes \mathbf{Q}_p$, I don't know if it is $\Gamma$-equivariant)

(How) can I conclude $\dim \mathrm{H}^2_{fppf}(X,\mathbf{Q}_p(1)) = \rho(X)$ for $X$ a product of smooth projective curves and Abelian varieties from this?

So the question is:

Do we have $\dim \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1))^\Gamma = \dim \mathrm{H}^2_{et}(X,\mathbf{Q}_\ell(1)) = \dim (\mathrm{H}^2_{cris}(\bar{X}/W) \otimes \mathbf{Q}_p)^\Gamma$

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    $\begingroup$ Tate's conjecture implies that the Frobenius action on the cohomology is semi-simple, hence LHS is equal to the multiplicity of 1 - X as a factor of the char poly of Frobenius, which is indep of $\ell$. This works for crystalline cohomology, which is the "usual" choice for $\ell = p$. I know nothing about fppf cohomology but maybe something similar works for that too? $\endgroup$ – David Loeffler Feb 15 '17 at 7:35
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    $\begingroup$ Here's a suggestion. The RHS is at least the LHS (that's clear). Moreover, the LHS can also be computed via crys coh as David Loeffler says. Now, fppf cohomology injects into crys cohomology, so it seems that this way you might be able to deduce that the RHS is at most the LHS. Could this work? $\endgroup$ – Ariyan Javanpeykar Feb 15 '17 at 7:42
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    $\begingroup$ @TimoKeller This is in Section 5 of Illusie's paper on crystalline cohomology. $\endgroup$ – Ariyan Javanpeykar Feb 15 '17 at 15:27
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    $\begingroup$ @TimoKeller Sorry, I wrote that a bit too hastily. It's Complexe de de Rham-Witt et cohomologie crystalline. Have a look at Theoreme 5.14, for example. $\endgroup$ – Ariyan Javanpeykar Feb 15 '17 at 17:04
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    $\begingroup$ @TimoKeller David's argument works because of the cohomological proof of the Weil conjectures, which implies the independence of the characteristic polynomial. $\endgroup$ – Will Sawin Jul 7 '17 at 10:03

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