On the $p$-adic cohomology  of the Lubin-Tate tower

Scholze, Peter

doi:10.24033/asens.2367

On the

p

-adic cohomology of the Lubin-Tate tower
[Sur la cohomologie

p

-adique de la tour de Lubin-Tate]

Scholze, Peter

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 811-863.

Résumé
Abstract

Nous prouvons un résultat de finitude pour la cohomologie $p$ -adique de la tour de Lubin-Tate. Pour tout $n \geq 1$ et corps $p$ -adique $F$ , cela fournit un functor canonique à partir de représentations $p$ -adiques admissibles de ${GL}_{n} (F)$ vers des représentations $p$ -adiques admissibles de ${Gal}_{F} \times D^{\times}$ , où ${Gal}_{F}$ est le groupe de Galois absolu de $F$ , et $D / F$ est l'algèbre à division centrale d'invariant $1 / n$ .

De plus, nous vérifions une compatibilité locale-globale pour cette correspondance, et une compatibilité avec le patching de Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.

We prove a finiteness result for the $p$ -adic cohomology of the Lubin-Tate tower. For any $n \geq 1$ and $p$ -adic field $F$ , this provides a canonical functor from admissible $p$ -adic representations of ${GL}_{n} (F)$ towards admissible $p$ -adic representations of ${Gal}_{F} \times D^{\times}$ , where ${Gal}_{F}$ is the absolute Galois group of $F$ , and $D / F$ is the central division algebra of invariant $1 / n$ .

Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.

MR Zbl

DOI : 10.24033/asens.2367

Classification : 14G22, 11S37, 11F80, 11F85.
Keywords: Lubin-Tate tower,

p

-adic cohomology, Langlands correspondence, patching, local-global compatibility, Shimura curves.
Mot clés : Tour de Lubin-Tate, cohomologie

p

-adique, correspondance de Langlands, patching, compatibilité local-global, courbes de Shimura.

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     author = {Scholze, Peter},
     title = {On the $p$-adic cohomology  of the {Lubin-Tate} tower},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {811--863},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {4},
     year = {2018},
     doi = {10.24033/asens.2367},
     mrnumber = {3861564},
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Scholze, Peter. On the $p$-adic cohomology  of the Lubin-Tate tower. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 811-863. doi : 10.24033/asens.2367. https://www.numdam.org/articles/10.24033/asens.2367/

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