Breuil–Mézard conjecture

Breuil–Mézard conjecture
Name	Breuil–Mézard conjecture
Field	Number theory
Introduced	1999
Authors	Christophe Breuil; Ariane Mézard
Related	Langlands program; Fontaine–Mazur conjecture; modularity theorem

Breuil–Mézard conjecture The Breuil–Mézard conjecture proposes a precise relationship between modular representation theory of GL_2 over local fields, arithmetic of local Galois representations, and the geometry of deformation rings arising in the study of Fermat's Last Theorem-era problems. It predicts that certain multiplicities appearing in the reduction modulo a prime of local algebraic representations correspond to Hilbert–Samuel multiplicities of local potentially semi-stable deformation rings tied to specific inertial types and Hodge–Tate weights. The conjecture sits at the crossroads of the Langlands program, p-adic Hodge theory, and the theory of modular forms.

Statement of the conjecture

The conjecture asserts that for a two-dimensional continuous representation of the absolute Galois group of a p-adic field, the mod-p multiplicities of Serre weights in the reduction of corresponding lattices in algebraic representations of GL_2 match the Hilbert–Samuel multiplicities of components of framed deformation rings parameterizing potentially semi-stable lifts with fixed inertial type and Hodge–Tate weights. It relates algebraic multiplicities in representations of GL_2 and arithmetic multiplicities on deformation rings constructed from Mazur-style deformation theory, incorporating notions from Fontaine-style classification such as potentially crystalline and potentially semi-stable conditions. Concretely, it produces equalities between linear combinations of Serre weight multiplicities and cycles on deformation spaces over completed local noetherian rings.

Historical background and motivation

The conjecture originated in work of Christophe Breuil and Ariane Mézard around 1999 amid developments following progress on the Taniyama–Shimura–Weil conjecture and the proof of Fermat's Last Theorem by Andrew Wiles and Richard Taylor. It evolved as part of efforts to understand the mod-p and p-adic local Langlands correspondences pursued by groups around Colmez, Emerton, Kisin, and Breuil. Motivations included clarifying Serre's conjectures refined by Jean-Pierre Serre and providing tools for modularity lifting results used by Kisin in proofs of potential modularity for elliptic curves over totally real fields and for the Fontaine–Mazur conjecture implications pursued by Calegari and Geraghty.

Connections with Galois representations and deformation rings

Central objects are two-dimensional representations of the absolute Galois group of a p-adic field, studied via p-adic Hodge theory classifications such as crystalline, semi-stable, and de Rham. Deformation rings introduced by Barry Mazur and developed by Richard Taylor and Mark Kisin parametrize lifts of residual representations and encode data about inertial types and Hodge–Tate weights. The conjecture links Serre weights—appearing in the mod-p representation theory of GL_2 and in the work of Serre and Edixhoven—to cycles on potentially semi-stable deformation rings studied using techniques from commutative algebra of local noetherian rings, Hilbert–Samuel multiplicity theory, and the geometry of Rapoport–Zink spaces and Shimura varieties investigated by Harris, Taylor, and Lan.

Known cases and proofs

Progress includes proofs in many cases by Mark Kisin using patching and Taylor–Wiles–Kisin methods, work of Toby Gee and collaborators extending to unitary groups and potentially Barsotti–Tate cases, and results by David Savitt and F. Diamond on explicit weight calculations. The conjecture is established for many two-dimensional representations over unramified extensions of Q_p and for certain inertial types via methods of Emerton and Gee, while general cases remain open. Key milestones involved linking the conjecture with the Breuil–Mézard machinery in papers by Kisin, Shotton, and Paškūnas who exploited the p-adic local Langlands correspondence and detailed study of blocks in mod-p representation theory of GL_2.

Applications and consequences

When available, the conjecture has powerful consequences for modularity lifting theorems of Wiles-type, refinements of Serre weight conjectures for modular forms, and classification results in the p-adic local Langlands program pursued by Colmez and Breuil. It informs the structure of Hecke algebras acting on completed cohomology in the work of Emerton and the formulation of potential automorphy theorems used by Clozel and Harris. The predicted equalities of multiplicities provide input for level-raising and level-lowering arguments used in the proof of modularity of Galois representations attached to Hilbert modular forms and automorphic forms on unitary and symplectic groups.

Technical tools and methods used

Techniques include deformation-theoretic patching methods inspired by the Taylor–Wiles strategy, local model calculations employing Breuil–Kisin modules, explicit classification via (phi, Gamma)-modules over the Robba ring, and representation-theoretic analysis of mod-p blocks pioneered by Vignéras and Paškūnas. Algebraic geometry of formal schemes, computation of Hilbert–Samuel multiplicities, and use of Hecke operators in completed cohomology spaces feature prominently, as do comparisons with the p-adic local Langlands correspondence results of Colmez and global methods from Calegari–Geraghty style potential automorphy arguments.

Category:Number theory