General cohomology groups and motives

2011-11-23T22:51:17Z

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.

General cohomology groups and motives - MathOverflow

General cohomology groups and motives