Théorie de Lubin-Tate non abélienne l-entière

Abstract

For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformations of matching mod-l representations. Roughly speaking, we can decompose "the supercuspidal part" of the l-integral cohomology as a direct sum, indexed by irreducible supercuspidals πmod l, of tensor products of universal deformations of πand of its two mates. Besides, we also get a virtual realization of both the semi-simple Langlands-Vigneras correspondence and the l-modular Langlands-Jacquet transfer for all representations, by using the cohomology complex and working in a suitable Grothendieck group.

References in corpus (3)

• Theorie de Lubin-Tate non-abelienne et representations elliptiques
• On the local Langlands correspondence
• On $l$-adic families of cuspidal representations of $\GL_2(\Q_p)$

Cited by in corpus (15)

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• The local Jacquet--Langlands correspondence and congruences modulo {\ell}
• A congruence property of the local Langlands correspondence
• Supercuspidal representations of ${\mathrm{GL}}_n(F)$ distinguished by a Galois involution
• Blocks of the category of smooth $\ell$-modular representations of $GL(n,F)$ and its inner forms: reduction to level-$0$
• Potentially good reduction loci of Shimura varieties
• Compactly supported cohomology and nearby cycle cohomology of open Shimura varieties of PEL type
• Ihara's lemma and level rising in higher dimension
• On mod p non-abelian Lubin-Tate theory for GL_2(Q_p)
• Local Ihara's lemma and applications
• Eichler-Shimura relations for local Shimura varieties
• Sur les extensions intermédiaires des systèmes locaux d'Harris-Taylor
• A geometric interpretation of the nilpotent part of local Langlands correspondence modulo l