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Let $X$ be a variety over $\mathbb{Q}$. Let $\mathcal{F}$ be a sheaf on $X$. Then we have an action of $Gal(\mathbb{Q})$ on $H_{et}^i(X,\mathcal{F})$. In certain cases we can say a lot about this action. For example if $\mathcal{F}$ is the constant sheaf $\mathbb{Q}_ p$ for some prime $p$, or the constant sheaf $\mathbb{C}$. In those cases $H_{et}^*(\ \underline{},\mathcal{F})$ is a Weil cohomology; and so we conjecture that these representations come from motives, and in particular we have the Langlands conjectures about how these representations are ``nice'' (i.e. automorphic).

My question is: what can we say about the representation $H_{et}^i(X,\mathcal{F})$ for a general $\mathcal{F}$? What if $\mathcal{F}$ is, for example, not constant? Do we have an equivalent conjecture to the Langlands conjectures? (i.e. is there a generalization for a general $\mathcal{F}$ to the statement that $H_{et}^i(X,\mathcal{F})$ should be automorphic?)

My humility when it comes to the Langlands conjectures behooved me to put a community wiki stamp on this question on the off chance my question strikes experts as silly and/or vague.

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The setup here, strictly speaking, is not correct. The sheaves which give rise to etale cohomology are sheaves on the etale site of X, not on X itself. Furthermore, etale cohomology with $\mathbf{Q}_p$-coefficients is defined as the inverse limit of etale cohomology groups with coefficients in $\mathbf{Z}/p^i\mathbf{Z}$, tensored with $\mathbf{Q}_p$. I believe that $\mathbf{Q}_p$- or $\mathbf{C}$-coefficients yield groups which are usually identically zero. The analogue of "nonconstant sheaf" in this setting might be something like "lisse $\ell$-adic sheaf", but I am far from an expert. – David Hansen Nov 23 '11 at 23:44
Hmm... I was careless with my speech. You are right that I meant $\mathcal{F}$ as sheaf on the etale site on $X$ rather than $X$. (I viewed that as implicit by the fact that I look at $H_{et}^*(X,\mathcal{F})$.) Your other comment is more substantive -- is it true that $H^i_{et}(X,\mathbb{Q}_p)$ where $\mathbb{Q}_p$ is the constant sheaf on the etale site over $X$ is different from the inverse limit of the $H^i_{et}(X,\mathbb{Z}/p^i\mathbb{Z})$? – James D. Taylor Nov 23 '11 at 23:53
The computation of "naive etale cohomology" as you give it here is on page 118 of my copy of Freitag-Kiehl; as David Hansen states, with $\mathbb{Q}_p$ coefficients, all higher cohomology vanishes. The right thing here is not etale cohomology, but rather $\ell$-adic cohomology. FK also gives the naive computation for $\mathbb{Z}_p$ coefficients, which is pretty interesting, actually. One reason to think that this inverse limit gives the "right thing" is that one expects $H^1$ to be dual to the $\ell$-adic Tate module for an elliptic curve, and the naive definition doesn't work here, whereas – Daniel Litt Nov 23 '11 at 23:54
(cont.) $\ell$-adic definition does work. – Daniel Litt Nov 23 '11 at 23:55
I'm not sure if lisse l-adic sheaf is really what I would be looking for. Ideally, I would like a statement that would also make sense for $\mathcal{F}$ a linear algebraic group scheme (which would restrict us to non-abelian cohomology), although that might be asking too much. For now it would suffice to restrict ourselves to sheaves into abelian groups. – James D. Taylor Nov 23 '11 at 23:56

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