vanishing of automorphic bundles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:00:52Z http://mathoverflow.net/feeds/question/66060 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66060/vanishing-of-automorphic-bundles vanishing of automorphic bundles Przemyslaw Chojecki 2011-05-26T14:49:42Z 2011-05-26T14:49:42Z <p>Let $S _K = S_K(G,X)$ be a Shimura variety of dimension $n$. Let $\xi$ be a (finite-dimensional) representation of $G$, which gives rise (by a construction of Harris) to an automorphic bundle $V(\xi)$ on $S_K$. </p> <p>What is known about the vanishing of $H^i(S_K, V(\xi))$? </p> <p>What I would like to know ideally, is whether we have (for example, for compact Shimura varieties of PEL-type) $H^i(S_K, V(\xi)) = 0$ for each $i \not = n$? (possibly, after a localisation at some "non-Eisenstein" ideal). Could we expect it in general?</p> <p>I am aware of results of Lan-Suh and Mokrane-Tilouine (who proved this result in some special cases under certain conditions on $\xi$), though I still have not digested it properly. Any references and comments are welcome!</p>