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Zbl 1138.11329
Hausberger, Thomas
Uniformization of Laumon-Rapoport-Stuhler varieties and Drinfeld-Carayol conjecture. (Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol.) (French)
[J] Ann. Inst. Fourier 55, No. 4, 1285-1371 (2005). ISSN 0373-0956; ISSN 1777-5310/e

Summary: Let us consider Laumon-Rapoport-Stuhler modular varieties $Ell$ for ``D- elliptic sheaves", which are defined over a function field $F$ in one variable over a finite field, for a division algebra $D$ of dimension $d^2$ over $F$. We show that these varieties admit, at a place $o$ of $F$ where $D_o$ is a skew field of invariant $1/d$, a rigid-analytic uniformization by Drinfeld's space $\Omega^d$, or by the coverings $\Sigma_n^d$ of $\Omega^d$ (depending on the level structure). This result is the analogue of Čerednik's theorem, which is well known in the number field case. As an application, we prove a conjecture of Carayol's: the inductive limit $\Psi_d$ over $n$ of the $\ell$-adic cohomology groups with support, in median degree, of the coverings $\Sigma_n^d$ - on which the product $\text{GL}_d (F_o)\times D_o^* \times W_{F_o}$ acts - yields a geometrical simultaneous realization of the local Langlands and Jacquet-Langlands correspondences. Our proof is of ``global" nature: using the uniformization theorem, we compare the local representation $\Psi^d$ to the global cohomology of the moduli variety $Ell$.

MSC 2000:: *11G18 Arithmetic aspects of modular and Shimura varieties
11S37 Langlands-Weil conjectures, nonabelian class field theory
11G09 Drinfel'd modules, etc.
14G22 Rigid analytic geometry

Keywords: Rigid-analytic uniformization; Drinfeld modular varieties; local Langlands correspondence

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