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Laurent Fargues

Publications and source records attributed to Laurent Fargues.

13 recordsLinked to original sources
arXiv2024

Geometrization of the local Langlands correspondence

Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of $L$-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define $L$-parameters associated with irreducible smooth representations of $G(E)$, a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of $L$-parameters on the category of $\ell$-adic sheaves on $\mathrm{Bun}_G$.

math.RT↗
arXiv2019

Théorie de la réduction pour les groupes p-divisibles

Starting from our work on Harder-Narasimhan filtrations of finite flat group schemes over a $p$-adic field, we developp a theory of Harder-Narasimhan filtrations for $p$-divisible groups. We apply this to the study of the geometry of period morphisms for Rapoport-Zink spaces and to the $p$-adic geometry of Shimura varieties. We define and study in particular some fundamental domains for the action of Hecke correspondences.

math.NT↗
arXiv2019

Groupes analytiques rigides p-divisibles II

Let $K$ be a $p$-adic field. We continue to develop the theory of rigid analytic $p$-divisible groups over $K$. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic $p$-divisible groups "in finite level" without perfectoid spaces. We then establish some results about families of rigid analytic $p$-divisible groups. This allows us to prove a "minimality" result in the sense of birationnal geometry for integral models of unramified Rapoport-Zink spaces.

math.AG↗
arXiv2017

On the structure of some p-adic period domains

We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge-Newton decomposable.

math.AG↗
arXiv2017

Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

We give a geometric Langlands type proof of the geometrization conjecture of the local Langlands correspondence introduced by the author for GL_1. For this we study an Abel-Jacobi morphism. We prove that this morphism is a pro-étale locally trivial fibration in simply connected diamonds in high degree. Those diamonds are absolute punctured Banach-Colmez spaces that we study in details.

math.AG↗
arXiv2006

Comparaison de la cohomologie des tours de Lubin-Tate et de Drinfeld et correspondance de Jacquet-Langlands geometrique

This article is the last one about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the existence of an isomorphism between the compactly supported etale cohomology of the Lubin-Tate and Drinfeld towers, and more generally their equivariant cohomology complex. We also prove the existence of a geometric local Jacquet-Langlands correspondence between some equivariant rigid etale sheaves on Gross-Hopkins period space $\mathbb{P}^{n-1}$ and Drinfeld one $Ω$.

math.NT↗
arXiv2006

L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld : demonstration du resultat principal

This is the fourth article about the isomorphism between Lubin-Tate and Drinfeld towers. We prove the final result concerning the isomorphism that is to say the existence of an equivariant isomorphism between some blow-up of the formal scheme associated to Lubin-Tate space with infinite level constructed cellularly before and a blow-up of another formal scheme associated to Drinfeld tower. We follow the strategy of Gerd Faltings.

math.NT↗
arXiv2006

Application de Hodge-Tate duale d'un groupe de Lubin-Tate, immeuble de Bruhat-Tits du groupe lineaire et filtrations de ramification

One of the goals of this article is to describe the isomorphism between Lubin-Tate and Drinfeld towers at the level of their skeletons after taking quotient by $\GL_n (Ø_F)\times Ø_D^\times$ or $I\timesØ_D^\times$ where $Ø_D$ is the maximal order in the division algebra with invariant $\frac{1}{n}$ over $F$ and $I$ a Iwahori subgroup of $\GL_n$. We give applications to the theory of canonical subgroups on Lubin-Tate spaces, the description of spherical Hecke orbits in those spaces, fundamental domains for Hecke correspondences and the Gross-Hopkins period mapping. We also study in details the ramification filtrations (upper and lower) and the Hodge-Tate map of a one dimensional formal $p$-divisible group.

math.NT↗
arXiv2005

Filtration de monodromie et cycles evanescents formels

V.Berkovich, K.Fujiwara and R.Huber have proved independently by different methods that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by P.Boyer in his work on some "simple Shimura varieties".

math.AG↗
arXiv2001

Correspondances de Langlands locales dans la cohomologie des espaces de Rapoport-Zink

We generalize the work of M. Harris and R. Taylor on the local Langlands correspondence for the linear group over $\mathbb{Q}_p$. We prove some cases of the Kottwitz conjectures for the supercuspidal part of the compactly supported $\ell$-adic cohomology of Rapoport-Zink rigid-analytic spaces. This means that we decompose this part in terms of local Langlands correspondences. In particular, we prove the first case of a non-abelian reciprocity law constructed geometrically and associated to groups other than inner forms of the linear group. More precisely, we do this for the unramified unitary group in three variables over $\mathbb{Q}_p$.

math.NT↗