On the vanishing of the measurable Schur cohomology groups of Weil groups

2002/09/18 by C. S. Rajan, Rajan, C. S.
Mathematics · #11R34 (Primary) 22E55 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.NT #msc:11R34 #msc:22E55

paper · pdf · doi:10.48550/arxiv.math/0209219

To appear in Compositio Mathematica. Minor editing, and a conjecture in the earlier version on the vanishing of the higher cohomology groups of Weil groups is false

openalex publication_date 2002/09/18 · arxiv created 2003/02/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in \C^* (or more generally, with coefficients in the complex points of a tori over \C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse proving that admissible homomorphisms of a Weil group to the Langlands dual of a reductive group can be lifted to an extension of the Langlands dual group by a tori.

Related

• On discontinuities of cocycles in cohomology theories for topological groups
• On Cohomology theory for topological groups
• Vanishing of the second L^p-cohomology group for most semisimple groups of rank at least 3
• A remark on the Langlands correspondence for tori
• Cohomology of p-adic fields and Local class field theory