Fargues Scholze Local Langlands. This is a review of the work of the authors on the geometrization

This is a review of the work of the authors on the geometrization of the local Langlands correspondence. We prove that V. Introduction I. In particular, we define a category of-adic sheaves on the stack Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from ℓ -adic sheaves to motivic sheaves. Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from ℓ -adic sheaves to motivic sheaves. Until further notice, we will simplify our life by assuming that G is spl We develop an analogue of Fargues’ geometrization of the local Langlands correspon-dence in the case of real groups. This includes a new formalism of G(R)-representations and a new In particular, we define a category of-adic sheaves on the stack BunG of G-bundles on the Fargues-Fontaine curve, prove a geometric Satake equivalence over the Fargues-Fontaine This led Fargues to speculate [2] that it should be possible to interpret local arithmetic Langlands at nonarchimedean places as geometric Langlands on an “exotic curve,” the Laurent Fargues and Peter Scholze Abstract. Lafforgue’s global Langlands correspondence is compatible with Fargues–Scholze’s semisimplified local Langlands corresp n-dence. Roughly speaking, the 1. This says there is a compatibility between our conjectural local perverse sheaf F' and Caraiani-Scholze perverse Given a prime p, a finite extension L/Qp, a connected p -adic reductive group G/L, and a smooth irreducible representation π of G(L), Fargues-Scholze recently attached a holze on Fargues’s geometric local Langlands program. Introduction The purpose of this paper is to explain about Fargues–Scholze’s monumental paper [24] on the geometrization of the local Langlands correspondence, which implements Laurent Fargues and Peter Scholze have found a new, more powerful way of connecting number theory and geometry as part of the The aim of this seminar is to study an analog of the Fargues{Scholze conjecture with Qp replaced by the archimedean local eld R and with the Fargues{Fontaine curve replaced by the twistor-P Can we show that the Fargues-Scholze Local Langlands correspondence is compatible with other instances of the correspondence? Namely, given a local Langlands correspondence: The purpose of this paper is to explain about Fargues–Scholze’s monumental paper [24]on the geometrization of the local Langlands correspondence, which implements and advances the Abstract. The local Langlands correspondence reducible smooth representa-tions of G(E), for a reductive group G over a local eld E. We explain the geometry of the stack Following the idea of [Far16], we develop the foundations for the geometric Langlands program on the Fargues-Fontaine curve. As a consequence, we . The generic ber of We prove that Fargues–Scholze's semisimplified local Langlands correspondence (for quasisplit groups) with Fℓ -coefficients is compatible This compatibility result is then combined with the spectral action constructed by Fargues and Scholze (2021, Chapter X), to verify their categorical form of the local Langlands In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues. Until further notice, we will simplify our life by The goal of Fargues–Scholze is to propose a uniform local Langlands correspondence, constructing a map π 7→φπ, which comes out of some more powerful structure and which we In this article we give an exposition of a conjecture at the frontier be-tween p-adic Hodge theory, the geometric Langlands program and the local Langlands correspondence. 1. The local Langlands correspondence ible smooth representa-tions π of G(E), for a reductive group G over a local field E. In particular, I. Following the idea of [Far16], we develop the foundations of the geometric Langlands program A key insight of Harris and Taylor was to geometrically realize the Local Langlands correspondence for G = GLn in the cohomology of Rapoport-Zink spaces. In particular, we define a category of l-adic sheaves V5A5: Geometrization of the local Langlands correspondence The goal of this course is to give an overview of the approach of Laurent Fargues and myself on the local Langlands Geometrization of the local Langlands correspondence Laurent Fargues and Peter Scholze . We extend our methods from [24] to reprove the Local Langlands Corre-spondence for GLn over p-adic elds as well as the existence of `-adic Galois represen-tations attached to bstract. Let E be a local field whose residue field has characteristic p, and let G be a connected reductive group over E. I can't even concisely state the In section 7 we explain the local/global compatibility part of the conjecture. This is an archimedean version of the Fargues-Scholze work on local Langlands at non-archimedean primes which uses ideas of geometric Langlands, but on the Fargues Geometriz Laurent Fargues and Peter Scholze op the foundations of the geometric Langlands program on the Fargues–Fontaine curve.

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