Galois representations arising from some compact Shimura varieties

Our aim is to establish some new cases of the global Langlands correspondence for GL(m). A long the way we obtain a new result on the description of the cohomology of some compact Shimura varieties. Let F be a CM field with complex conjugation c and II be a cuspidal automorphic representation of GL(m)(A(F)). Suppose that II similar to II o c and that II infinity is cohomological. A very mild conditionon II infinity is imposed if m is even. We prove that for each prime l there exists a continuous semi simple represent at ion R-l(II):Gal((F) over bar =F)-> GL(m)((Q) over bar (l)) such that II and R-l(II) correspond via the local Langlands correspondence (established by Harris-Taylor and Henniart) at every finite place w-l of F (\local-global compatibility"). We also obtain several additional properties of R-l(II) and prove the Ramanujan-Petersson conjecture for II. This improves the previous results obtained by Clozel, Kottwitz, Harris-Taylor and Taylor-Yoshida, where it was assumed in addition that II is square integrable at a finite place. It is worth noting that the mild condition on II infinity in our theorem is removed by a p-adic deformation argument, thanks to Chenevier-Harris Our approach generalizes that of Harris-Taylor, which constructs Galois representations by studying the l-adic cohomology and bad reduction of certain compact Shimura varieties attached to unitary similitude groups . The central part of our work is the computation of the cohomology of the so-called Igusa varieties. Some of the main tools are the stabilized counting point formula for Igusa varieties and techniques in the stable and twisted trace formulas. Recently there have been results by Moreland Clozel-Harris-Labesse in a similar direction as ours. Our result is stronger in a few aspects. Most notably, we obtain information about R-l(II) at ramified places.