2019
DOI: 10.1090/jams/935
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Cohomologie 𝑝-adique de la tour de Drinfeld: le cas de la dimension 1

Abstract: We compute the p p -adic geometric Ă©tale cohomology of the coverings of the Drinfeld half-plane, and we show that, if the base field is Q p \mathbf {Q}_p , this cohomology encodes the p p -adic local Langlands correspondence for 2 2 -dimensional de Rham representations (of weight 0 0 and 1 1 ).

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Cited by 16 publications
(53 citation statements)
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“…(ii) We have similar results [21] for Ă©tale coverings of ℩ K . This is used to show that, for K = Q p , the p-adic Ă©tale cohomology of these coverings encodes a part of the p-adic local Langlands correspondence, which yields a geometric realization for this correspondence (it is a classical result that the -adic Ă©tale cohomology of these coverings can be used to provide a geometric realization of the classical local Langlands correspondence).…”
Section: Stein Rigid Analytic Varietiessupporting
confidence: 81%
“…(ii) We have similar results [21] for Ă©tale coverings of ℩ K . This is used to show that, for K = Q p , the p-adic Ă©tale cohomology of these coverings encodes a part of the p-adic local Langlands correspondence, which yields a geometric realization for this correspondence (it is a classical result that the -adic Ă©tale cohomology of these coverings can be used to provide a geometric realization of the classical local Langlands correspondence).…”
Section: Stein Rigid Analytic Varietiessupporting
confidence: 81%
“…Il reste donc Ă  dĂ©montrer le a), et il suffit de le faire pour k = 0. Si n = 0, il s'agit d'un exercice amusant (25) laissĂ© au lecteur. Dans le cas gĂ©nĂ©ral, on utilise le fait que ÎŁ n est un revĂȘtement Ă©tale de ÎŁ 0 .…”
Section: Numérologie Et Lissité De Hunclassified
“…mais cela demande plus de travail : l'argument global utilisĂ© permet de dĂ©montrer facilement qu'il n'y a pas d'autre reprĂ©sentation de la sĂ©rie discrĂšte (ainsi que beaucoup de sĂ©ries principales) parmi les sous-quotients de H 1 dR,c (ÎŁ n ) ρ √ , mais il ne semble pas facile d'exclure la prĂ©sence de n'importe quelle sĂ©rie principale avec ce genre d'argument (dans le cas ℓ-adique, ce genre de difficultĂ© est contournĂ© en utilisant l'isomorphisme de Faltings-Fargues [36,37] ; cela semble plus dĂ©licat dans notre situation, mais c'est effectivement ce qui est fait dans [25], oĂč le rĂ©sultat est dĂ©montrĂ© avec GL 2 (Q p ) remplacĂ© par GL 2 (F ), avec F une extension finie quelconque de Q p ). On suppose que K p = ℓ =p K ℓ , oĂč K ℓ est un sous-groupe ouvert compact deB * (Q ℓ ), et on suppose qu'il existe au moins un premier ℓ 0 tel que K ℓ0 soit sans torsion.…”
Section: Uniformisation P-adique Et Cohomologie De De Rhamunclassified
See 1 more Smart Citation
“…Despite these difficulties, recently Colmez-Dospinescu-Nizio l, [CDN20a], were able to show that the geometric p-adic Ă©tale cohomology of the Drinfeld tower over Q p in dimension 1 realizes the p-adic local Langlands correspondence for GL 2 (Q p ), for the 2-dimensional de Rham representations of G Qp of Hodge-Tate weight 0 and 1, whose associated Weil-Deligne representation is irreducible. Moreover, their computation suggests that the geometric p-adic Ă©tale cohomology of the Drinfeld tower over K in dimension 1 should encode a still hypothetical p-adic Langlands correspondence for GL 2 (K), for a general K.…”
Section: Introductionmentioning
confidence: 99%