Is H^2(W_p,C^times) well-known? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:32:43Z http://mathoverflow.net/feeds/question/52270 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52270/is-h2w-p-ctimes-well-known Is H^2(W_p,C^times) well-known? David Hansen 2011-01-16T23:12:15Z 2011-01-17T01:12:19Z <p>Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated?</p> <p>(This group comes up, at least for me, when you want to compare two Weil group representations whose projectivizations agree.)</p> http://mathoverflow.net/questions/52270/is-h2w-p-ctimes-well-known/52278#52278 Answer by Marty for Is H^2(W_p,C^times) well-known? Marty 2011-01-17T01:12:19Z 2011-01-17T01:12:19Z <p>It is known that $H^2(W, C^\times)$ is trivial, when $W$ is the Weil group of a global or local field, with the trivial action on $C^\times$, and the cohomology is taken in the sense of Moore (measurable cochains). This is the main result of C.S. Rajan, "On the vanishing of the measurable cohomology groups of Weil groups", Compositio Math. 140 (2004) 84-98 (also easy to find online). </p> <p>Is this well-known? I don't know. I only found this paper recently, and hopefully now it will become better-known!</p>