Reference for Unitary Group attached to \$E/k\$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:49:56Z http://mathoverflow.net/feeds/question/20339 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20339/reference-for-unitary-group-attached-to-e-k Reference for Unitary Group attached to \$E/k\$ Dipramit Majumdar 2010-04-04T22:55:13Z 2010-04-05T07:05:15Z <p>Unitary groups are very important objects in the setting of Langland's Conjecture because of the existence of Shimura Variety ( which I don't know) and also because people know how to attach a galois representation to a automorphic representation of unitary groups in almost all cases (By the work of Taylor, Harris and many others)(Which also I don't know).</p> <p>I was trying to learn about unitary groups attached to \$(k,E,D,*)\$ where say \$k\$ is a totally real field,\$E\$ CM field of deg 2 over \$k\$, \$D\$ is a central simple algebra of rank \$n^2\$ over E, and \$*\$ \$k\$ algebra anti-involution of 2nd kind on \$D\$ (i.e. coinciding with the action of non-trivial element of \$Gal(E/k)\$ on \$E\$).</p> <p>But I could not really find a reference for this. Essentially some authors define it a inner form of a particular quasi-split unitary group, and some authors define it as functor of points. Also it is commented that there is some sort of Global-Local patching going on. Can any one give me a reference where unitary groups is covered in some what details rather than a overview in 2 pages?</p> http://mathoverflow.net/questions/20339/reference-for-unitary-group-attached-to-e-k/20340#20340 Answer by Matthew Stover for Reference for Unitary Group attached to \$E/k\$ Matthew Stover 2010-04-04T23:12:57Z 2010-04-04T23:12:57Z <p>Scharlau's book, <em>Quadratic and hermitian forms</em>, gives the complete classification in Chapter 10.</p> http://mathoverflow.net/questions/20339/reference-for-unitary-group-attached-to-e-k/20368#20368 Answer by Paul Broussous for Reference for Unitary Group attached to \$E/k\$ Paul Broussous 2010-04-05T07:05:15Z 2010-04-05T07:05:15Z <p>For discussions on forms of classical groups you can look at:</p> <p>-- André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. 24 (1961), 589-623 (also in Oeuvres Complètes).</p> <p>-- Platonov-Rapinchuk, Algebraic groups and number theory, Ac. Press, 1994.</p> <p>-- The book of involutions, AMS Coll. Publ., vol. 44, 1998.</p> <p>-- Kneser, Lecture on Galois cohomology of classical groups, Tata Inst. of Fund. Research, Bombay, 1969.</p> <p>I've written a summary of Weil's theorems and proofs (following Platonov-Rapinchuk) in </p> <pre><code> http://www-math.univ-poitiers.fr/~broussou/formesgc.pdf </code></pre> <p>It is in French ... and only deals with local base fields. </p>