Paper · Feb 2, 2016

Geometrization of the local Langlands correspondence: an overview

Laurent Fargues

math.NTarXiv:1602.00999
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Abstract

This article is an overview of the geometrization conjecture for the local Langlands correspondence formulated by the author.

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Submitted
Feb 2, 2016
Subjects
math.NT, math.AG, math.RT
Geometrization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local (Laurent Fargues, 2017)1Fargues 2017 Etale cohomology of diamonds (Peter Scholze, 2017)2Scholze 2017 Higgs bundles and geometric structures on manifolds (Daniele Alessandrini, 2018)7Alessandrini 2018 Uniform models and short curves for random 3-manifolds (Peter Feller, Alessandro Sisto and Gabriele Viaggi, 2019)8Feller 2019 Canonical geometrization of orientable 3-manifolds defined by vector-colourings of 3-polytopes (Nikolai Erokhovets, 2020)4Erokhovets 2020 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)3de Freitas 2022 Geometrisation of 3-manifolds (Bruno Martelli, 2026)6Martelli 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)5Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

    Laurent Fargues

    Topics Geometrization · Same author Laurent Fargues · Shared terms “Langlands”, “correspondence”, “geometrization conjecture”

  2. Etale cohomology of diamonds

    Peter Scholze

    Topics Geometrization · Shared terms “Langlands correspondence”, “local Langlands”, “geometrization conjecture”

  3. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization

  4. Canonical geometrization of orientable -manifolds defined by vector-colourings of -polytopes

    Nikolai Erokhovets

    Topics Geometrization · Shared terms “geometrization conjecture”

  5. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “geometrization conjecture”

  6. Geometrisation of 3-manifolds

    Bruno Martelli

    Topics Geometrization · Shared terms “overview”

  7. Higgs bundles and geometric structures on manifolds

    Daniele Alessandrini

    Topics Geometrization · Shared terms “geometrization conjecture”

  8. Uniform models and short curves for random 3-manifolds

    Peter Feller, Alessandro Sisto and Gabriele Viaggi

    Topics Geometrization · Shared terms “geometrization conjecture”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.