Progrès récents en fonctorialité de Langlands
We investigate the vertical version of the Sato-Tate conjecture for some GL₂ automorphic representations over totally real fields with specified local components at a finite set of finite places.
The “regular”trace formula, for a test function with a local component which is Iwahori-biinvariant and sufficiently regular with respect to the other components, is developed in the context of a reductive group. It is used to give a simple proof of the theory of base-change for cuspidal automorphic representations of which have a supercuspidal component. A purely local proof is given to transfer orbital integrals of sufficiently many spherical functions, by relating them to regular Iwahori functions....
Nous construisons un complexe de représentations localement analytiques de , associé à certaines représentations semi-stables de dimension du groupe de Galois absolu de . Nous montrons ensuite que l’on peut retrouver le -module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des -modules, de ce complexe dans le complexe de de Rham de l’espace de Drinfel’d de dimension . La preuve requiert le calcul de certains espaces de cohomologie localement...
We study the local factor at of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining their images under the Hecke algebra isomorphisms of Goldstein, Morris, and Roche.