Représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$

Schraen, Benjamin

Représentations localement analytiques de

{GL}_{3} (ℚ_{p})

Schraen, Benjamin

Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011), p. 43-145 / Harvested from Numdam

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Résumé
Abstract

Nous construisons un complexe de représentations localement analytiques de ${GL}_{3} (ℚ_{p})$ , associé à certaines représentations semi-stables de dimension $3$ du groupe de Galois absolu de $ℚ_{p}$ . Nous montrons ensuite que l’on peut retrouver le $(ϕ, N)$ -module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des $D ({GL}_{3} (ℚ_{p}))$ -modules, de ce complexe dans le complexe de de Rham de l’espace de Drinfel’d de dimension $2$ . La preuve requiert le calcul de certains espaces de cohomologie localement analytiques de sous-groupes unipotents à coefficients dans des séries principales localement analytiques.

We construct a complex of locally analytic representations of ${GL}_{3} (ℚ_{p})$ , which is associated to some semi-stable $3$ -dimensional representations of the absolute Galois group of $ℚ_{p}$ . Then we show that we can retrieve the $(ϕ, N)$ -filtered module of the Galois representation in the space of morphisms, in the derived category of $D ({GL}_{3} (ℚ_{p}))$ -modules, of this complex in the de Rham-complex of the $2$ -dimensional Drinfel’d’s space. For the proof, we compute some spaces of locally analytic cohomology of unipotent subgroups with coefficients in some locally analytic principal series.

Publié le : 2011-01-01

MR 2760195 | Zbl 1235.11108

DOI : https://doi.org/10.24033/asens.2140
Classification: 11F70, 11S20, 11S37, 11S80, 14G22, 22E50
Mots clés: correspondance de Langlands

p

-adique, espaces de Drinfel’d, représentations localement analytiques

p

-adiques

@article{ASENS_2011_4_44_1_43_0,
     author = {Schraen, Benjamin},
     title = {Repr\'esentations localement analytiques de $\mathrm {GL}\_3(\mathbb {Q}\_{p})$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {44},
     year = {2011},
     pages = {43-145},
     doi = {10.24033/asens.2140},
     mrnumber = {2760195},
     zbl = {1235.11108},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/ASENS_2011_4_44_1_43_0}
}

Schraen, Benjamin. Représentations localement analytiques de $\mathrm {GL}_3(\mathbb {Q}_{p})$. Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011) pp. 43-145. doi : 10.24033/asens.2140. http://gdmltest.u-ga.fr/item/ASENS_2011_4_44_1_43_0/

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