General cohomology groups and motives - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:12:15Z http://mathoverflow.net/feeds/question/81759 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81759/general-cohomology-groups-and-motives General cohomology groups and motives James D. Taylor 2011-11-23T22:51:17Z 2011-11-23T22:51:17Z <p>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).</p> <p>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?)</p> <p>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.</p>