Correspondance de Langlands géométrique pour les corps de fonctions. (Geometric Langlands correspondence for function fields)
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 an open subset \(S^ 0_ n\) of \(Fib_{X,n}\), whose fibres are projective spaces. The complex \(\tilde K_ E\) is supposed to satisfy certain properties formulated in conjecture (4.3.3) and (5.3.1) of the paper, which in turn would imply Langlands' conjecture. The most important conjectural property (modular invariance) is that \(\tilde K_ E\) should be the pullback of a perverse sheaf \(K_ E\) on \(Fib_{X,n}\). The latter should define the cusp form. As already mentioned the constant field need not be finite. In case of constant fields of characteristic 0 the authors gives some evidence for the conjectures by considering the characteristic varieties of the perverse sheaves involved.
Langlands conjecture on nonabelian class field theory
scientific article; zbMATH DE number 4081688
scientific article