p-adic Langlands correspondence

p-adic Langlands correspondence
Name	p-adic Langlands correspondence
Field	Number theory
Introduced	1990s
Notable persons	Pierre Deligne, Jean-Pierre Serre, Robert Langlands, Christophe Breuil, Colmez, Richard Taylor, Andrew Wiles, Barry Mazur, Mark Kisin

p-adic Langlands correspondence is a conjectural and partly established network of relationships between p-adic representations arising from Galois representations and p-adic analytic representations of p-adic reductive groups, extending the scope of the classical Langlands program and interacting with the theories of p-adic Hodge theory, modular forms, automorphic forms, and Iwasawa theory. It refines links found by Robert Langlands, Pierre Deligne, and Jean-Pierre Serre and has driven developments by Christophe Breuil, Colmez, Richard Taylor, and Barry Mazur among others. The correspondence sits at the crossroads of work on Shimura variety, moduli space, Galois deformation theory, and p-adic L-function constructions.

Introduction

The p-adic Langlands program seeks an equivalence or dictionary connecting continuous p-adic representations of the absolute Galois group of local fields and p-adic Banach space representations of groups such as GL_2(Q_p), with parallels to global conjectures linking p-adic automorphic forms on GL_n or unitary groups to p-adic Galois representations. Motivated by breakthroughs of Andrew Wiles on Taniyama–Shimura and modularity results of Richard Taylor and Frederick Diamond, the theory integrates techniques from p-adic Hodge theory, (φ,Γ)-module theory developed by Gerritzen, and deformation theory introduced by Barry Mazur.

Historical Development and Key Conjectures

Early clues emerged from connections between elliptic curves studied by Andrew Wiles and Shimura curves and the classical Langlands reciprocity envisioned by Robert Langlands. In the 1990s and 2000s, work of Christophe Breuil, Colmez, Mark Kisin, and Barry Mazur formulated precise local conjectures for GL_2(Q_p), while global conjectures were shaped by insights from Richard Taylor, Michael Harris, Laurent Clozel, and Edward Witten's influence on geometric perspectives. Major conjectures posit a functorial bridge between families of p-adic automorphic representations on unitary groups, Hilbert modular forms, and Siegel modular forms and p-adic families of Galois representations controlled by Hida family and Coleman family constructions.

Local p-adic Langlands Correspondence

The local theory centers on correspondences for groups such as GL_2(Q_p), with foundational constructions by Christophe Breuil and Pierre Colmez producing a correspondence between two-dimensional p-adic Galois representations and unitary Banach space representations of GL_2(Q_p). Techniques draw on (φ,Γ)-module theory of Jean-Marc Fontaine, Sen theory and classifications originating in Jean-Pierre Serre's work. Results for higher rank groups like GL_n remain conjectural, with partial advances by Mark Kisin, Vytautas Paškūnas, and Matthew Emerton linking completed cohomology of eigencurves to local p-adic representation theory.

Global p-adic Langlands Correspondence

Globally, the program aims to attach p-adic automorphic representations on reductive groups over number fields such as GL_n and certain unitary groups to p-adic Galois representations of the absolute Galois group of number fields. Constructions use completed cohomology introduced by Matthew Emerton, Eigenvariety technology pioneered by Fred Diamond and Robert Coleman, and p-adic interpolations of classical correspondences found in Langlands reciprocity. Global modularity lifting theorems from Andrew Wiles and Richard Taylor provide templates, while the arithmetic of Shimura varietys studied by Kisin and Harris supplies geometric input.

Methods and Constructions

Approaches employ p-adic analytic methods: (φ,Γ)-module classification from Jean-Marc Fontaine, Hodge–Tate theory, p-adic local Langlands techniques, and deformation theory of Galois representations as developed by Barry Mazur and Ken Ribet. Eigenvarieties and Coleman–Mazur eigencurve constructions allow p-adic variation as in work of Robert Coleman and Berthelot; Iwasawa theory and p-adic L-function methods supply arithmetic control. Categorical frameworks invoke derived category methods inspired by Alexander Beilinson and Joseph Bernstein, while categorical and geometric inputs draw on ideas from Geometric Langlands proponents such as Edward Frenkel.

Examples and Known Cases

The most complete case is the local correspondence for GL_2(Q_p), established by Christophe Breuil and Pierre Colmez, linking 2-dimensional p-adic representations coming from elliptic curves and modular forms to explicit unitary Banach representations. Global results include p-adic families attached to Hida familys and Coleman familys and modularity lifting theorems used in proofs by Andrew Wiles and Richard Taylor for Fermat-related cases. Partial higher-rank results appear in the work of Mark Kisin, Vytautas Paškūnas, and Matthew Emerton on GL_n-local and global compatibility and on the relationship with the cohomology of Shimura varietys.

Open Problems and Current Research

Active problems include formulating a comprehensive correspondence for GL_n over Q_p and general p-adic reductive groups, proving local-global compatibility in full generality, and connecting p-adic Langlands objects to the arithmetic of motives and non-abelian Iwasawa theory. Current research by groups associated with IHÉS, IAS, University of Cambridge, and Princeton University focuses on extending Colmez–Breuil methods, developing categorical p-adic techniques influenced by Geometric Langlands advances, and constructing p-adic L-functions in families. Contributions from researchers like Mark Kisin, Matthew Emerton, Vytautas Paškūnas, and Ludwig Fargues continue to shape progress toward broad conjectures in the p-adic realm.

Category:Number theory