local Langlands correspondence

local Langlands correspondence
Name	Local Langlands correspondence
Field	Number theory, Representation theory, Algebraic geometry
Introduced	1960s
Major figures	Robert Langlands, Pierre Deligne, Michael Harris, Richard Taylor, Andrew Wiles

local Langlands correspondence

The local Langlands correspondence is a conjectural and partly proven set of deep relationships between representations of reductive groups over local fields and Galois- or Weil–Deligne-type parameters, tying together the work of Robert Langlands, André Weil, John Tate, Pierre Deligne, and later developments by Michael Harris and Richard Taylor. It connects the representation theory of groups such as GL(n), SL(n), and other reductive groups over local non-archimedean fields like p-adic numbers () with arithmetic objects arising from the absolute Galois group of local fields, building on ideas from the Taniyama–Shimura conjecture and methods influenced by the Langlands program and the automorphic forms literature. The correspondence has driven progress in the proof of instances of the Artin conjecture, the proof of the local-global principle in many cases, and interactions with the theory of ℓ-adic representations, motives, and the trace formula.

Introduction

The correspondence was proposed in the wake of work by Robert Langlands and conceptualized through interactions with André Weil and John Tate; it aims to match irreducible admissible representations of reductive groups over local fields such as Q_p, R, and finite extensions thereof with continuous homomorphisms from the Weil group or Weil–Deligne group into complex or ℓ-adic Langlands dual groups like GL(n,C), SL(2,C), and general complex reductive groups introduced in the context of the Langlands dual group. This framework draws on tools and results from the Harish-Chandra theory of representations, the Grothendieck trace formula, and insights from the Bernstein center and Hecke algebra theory developed by authors such as I. N. Bernstein and H. Jacquet.

Statement of the Correspondence

For a non-archimedean local field F (for example Q_p or a finite extension) and a connected reductive group G over F like GL(n), the expected correspondence is a finite-to-one map between isomorphism classes of irreducible admissible representations of G(F) (built using smooth representations and parabolic induction as in works of Bernstein–Zelevinsky and Casselman–Shalika) and equivalence classes of L-parameters: continuous homomorphisms from the Weil–Deligne group W'_F into the L-group {}^LG (a semidirect product involving the Langlands dual group), satisfying local compatibility with L-factors and ε-factors defined by Igusa, Tate, and Deligne. Precise formulations require normalization choices from the Haar measure theory used in the Godement–Jacquet zeta integrals and compatibility with the local factors appearing in the Langlands–Shahidi method and the Rankin–Selberg convolution.

Construction and Methods

Constructions and proofs use a mix of global and local tools: the trace formula of James Arthur and Robert Kottwitz connects spectral data for adelic groups with orbital integrals, while techniques of base change and cyclic base change (pioneered by Langlands and Arthur–Clozel) reduce cases to GL(n). For GL(n), proofs by Michael Harris, Richard Taylor, Guy Henniart, and Colmez exploit global-to-local methods via the Taylor–Wiles method, modularity lifting theorems (related to Andrew Wiles and the proof of the Taniyama–Shimura–Weil conjecture), and properties of ℓ-adic Galois representations from the work of Pierre Deligne and Jean-Pierre Serre. Geometric methods use perverse sheaves, the geometric Langlands program influenced by Alexander Beilinson, Vladimir Drinfeld, and Edward Frenkel, and the theory of Bernstein blocks and types developed by Bushnell–Kutzko.

Examples and Known Cases

The correspondence is established for G = GL(1) via local class field theory of Artin reciprocity and local class field theory developed by Emil Artin and John Tate, and for G = GL(n) by results of Henniart, Harris–Taylor, and later refinements using techniques from p-adic Hodge theory by Colmez and Kisin. Work of Robert Langlands and James Arthur describes aspects for classical groups via the Arthur conjectures and recent progress by Mok and Arthur for unitary and orthogonal groups uses endoscopic classification developed with Clozel and Kottwitz. Explicit local correspondences are known for SL(2), U(n), and tamely ramified tori using methods by Bushnell–Henniart and DeBacker–Reeder.

Properties and Consequences

The correspondence preserves local invariants: matching of L-factors, ε-factors, and conductors as in the work of Deligne, Langlands–Shahidi, and Jacquet–Piatetski-Shapiro–Shalika. It organizes representations into L-packets and relates to the Satake isomorphism at unramified places, invoking the Satake transform and results by I. Satake and Cartan. Consequences include progress on the Artin conjecture for certain cases, compatibility with the local Langlands reciprocity for characters, and implications for the structure of automorphic representations appearing in the cohomology of Shimura varieties studied by Kottwitz, Harris–Taylor, and Lan–Taylor–Xiao.

Local-Global Compatibility

Local parameters arising from the global automorphic representation attached to an L-function should match the local components given by the local correspondence; this local-global compatibility is a cornerstone in proofs of modularity lifting results such as the Taylor–Wiles method and in the formulation of reciprocity laws connecting Galois representations constructed from Shimura varieties by Deligne and Kottwitz to automorphic data. Work by Buzzard, Gee, Calegari–Geraghty, and Barnet-Lamb extends local-global compatibility to p-adic families and potential automorphy, while the Fontaine–Mazur conjecture and results of Caraiani and Emerton relate p-adic Hodge-theoretic properties of local Galois representations to the local-automorphic side.

Category:Langlands program