#### Vol. 8, No. 3, 2014

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Lefschetz operator and local Langlands modulo $\ell$: the limit case

### Jean-François Dat

Vol. 8 (2014), No. 3, 729–766
##### Abstract

Let $K$ be a finite extension of ${ℚ}_{p}$ with residue field ${\mathbb{F}}_{q}$, and let $\ell$ be a prime such that $q\equiv 1\left(mod\phantom{\rule{1em}{0ex}}\ell \right)$. We investigate the cohomology of the Lubin–Tate towers of $K$ with coefficients in ${\overline{\mathbb{F}}}_{\ell }$, and we show how it encodes Vignéras’ Langlands correspondence for unipotent ${\overline{\mathbb{F}}}_{\ell }$-representations.

##### Keywords
local Langlands, Lefschetz operator, modulo $\ell$
##### Mathematical Subject Classification 2010
Primary: 11S37
Secondary: 11F70, 14G35
##### Milestones
Received: 18 June 2013
Revised: 7 November 2013
Accepted: 10 December 2013
Published: 31 May 2014
##### Authors
 Jean-François Dat Institut de Mathématiques de Jussieu Université Pierre et Marie Curie (Paris 6) 4, place Jussieu 75252 Paris France