Correspondance de Langlands géométrique pour les corps de fonctions. (Geometric Langlands correspondence for function fields)
type: Item
instance of: scholarly article
title: Correspondance de Langlands géométrique pour les corps de fonctions. (Geometric Langlands correspondence for function fields)
zbMATH Open document ID: 0662.12013
DOI: 10.1215/S0012-7094-87-05418-4
author: Gérard Laumon
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The paper tries to give a geometric interpretation of the Langlands conjecture on nonabelian class field theory in the case of function fields F of one variable over finite (or more general) constant fields. For simplicity only the simplest cases of geometrically irreducible, \(\overline{{\mathbb{Q}}}_{\ell}\) valued \(\ell\)-adic Galois representations E of \(Gal(\overline{F}:F)\) are considered, namely those which are completely unramified. According to the general conjecture to any such E should correspond a spherical cuspidal automorphic form on \(Gl(n,{\mathbb{A}}_ F)\), where \({\mathbb{A}}_ F\) is the ring of adeles attached to F and where n is the rank of E. These cusp forms are essentially determined by their eigenvalues under the Hecke operators and should be related to certain perverse sheaves \(K_ E\) on the moduli stack \(Fib_{X,n}=Gl(n,F)\setminus Gl(n,{\mathbb{A}}_ F)/K\) of rank n vector bundles over X, where K is maximal compact in \(Gl(n,{\mathbb{A}}_ F)\) and X is the smooth curve corresponding to F. More precisely the trace of Frobenius \(F_ x\) for \(x\in Fib_{X,n}\) on \(K_ E\) should give the value of the corresponding cusp form at the point x. The author proposes a procedure how to derive \(K_ E\) from E in purely geometrical terms. This is divided into two steps. First he derives from E a perverse sheaf \(W_ E\) on the moduli stack \(Coh_{X,{\mathcal O}}\) of coherent \({\mathcal O}_ X\) modules of finite length. The complex \(W_ E\) is the geometric analog of the Whittaker function attached to the conjectured cusp form. The construction of \(W_ E\) is related to Springer's construction of representations for Weyl groups. The second step is more subtle. It presents a geometric version of related work of Shalika on automorphic forms. By an iterative procedure a conjectural description of a certain perverse sheaf \(\tilde K_ E\) on \(U_ n\) is given, where \(U_ n\subset P^ 0_ n\) is an open subset of a space \(P^ 0_ n\to S^ 0_ n\) fibred over …
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 4081688
zbMATH Keywords: Langlands conjecture on nonabelian class field theory
zbMATH Keywords: function fields
zbMATH Keywords: spherical cuspidal automorphic form
zbMATH Keywords: ring of adeles
zbMATH Keywords: perverse sheaves
zbMATH Keywords: moduli stack
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4081688