Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations

Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt

TL;DR

This work develops a geometric framework for the Breuil–Mézard conjecture and the weight part of Serre's conjecture in dimension two by organizing moduli of mod $p$ Galois representations into stacks built from Breuil–Kisin modules and étale $\varphi$-modules. It introduces cycles $Z^{\sigma}$ on irreducible components labeled by Serre weights and proves that a representation lies in the weight set $W(\overline{r})$ precisely when it meets the support of the corresponding cycle, after passing through tamely potentially Barsotti–Tate deformation rings and related stacks $\mathcal{Z}^{\mathrm{dd}}$ and $\mathcal{Z}^{\tau}$. The paper establishes generic reducedness of the relevant deformation rings, provides a BM decomposition for the cycles on tamely BT stacks, and then transfers these results to the EGmoduli stack $\mathcal{X}_{2,\mathrm{red}}$, where non-Steinberg weights correspond to irreducible components and Steinberg weights decompose in a controlled way with multiplicity one. Collectively, these results realize a geometric Breuil–Mézard conjecture for the stacks $Z^{\mathrm{dd},1}$ and, via EGmoduli, for $X_{2,\mathrm{red}}$, offering a robust geometric lens on the $p$-adic Langlands program for $\mathrm{GL}_2$ over $p$-adic fields and advancing modularity-lifting techniques. The methods hinge on a blend of Breuil–Kisin module theory, $\varphi$-module techniques, versal deformation rings, and the interplay between local deformation data and global Serre weight information.

Abstract

We establish a geometrisation of the Breuil-Mézard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field.

The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations

TL;DR

This work develops a geometric framework for the Breuil–Mézard conjecture and the weight part of Serre's conjecture in dimension two by organizing moduli of mod Galois representations into stacks built from Breuil–Kisin modules and étale -modules. It introduces cycles on irreducible components labeled by Serre weights and proves that a representation lies in the weight set precisely when it meets the support of the corresponding cycle, after passing through tamely potentially Barsotti–Tate deformation rings and related stacks and . The paper establishes generic reducedness of the relevant deformation rings, provides a BM decomposition for the cycles on tamely BT stacks, and then transfers these results to the EGmoduli stack , where non-Steinberg weights correspond to irreducible components and Steinberg weights decompose in a controlled way with multiplicity one. Collectively, these results realize a geometric Breuil–Mézard conjecture for the stacks and, via EGmoduli, for , offering a robust geometric lens on the -adic Langlands program for over -adic fields and advancing modularity-lifting techniques. The methods hinge on a blend of Breuil–Kisin module theory, -module techniques, versal deformation rings, and the interplay between local deformation data and global Serre weight information.

Abstract

We establish a geometrisation of the Breuil-Mézard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field.
Paper Structure (8 sections, 25 theorems, 32 equations)

This paper contains 8 sections, 25 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Moduli stacks of Breuil--Kisin modules and étale phi-modules
  4. Generic reducedness of potentially Barsotti--Tate deformation ring
  5. A case of the classical geometric Breuil--Mézard Conjecture
  6. The geometric Breuil--Mézard conjecture for the stacks Z^ dd,1
  7. The geometric Breuil--Mézard conjecture for the stacks X_2,red
  8. A lemma on formal algebraic stacks

Key Result

Theorem 1.1

EGmoduli Each $\mathcal{X}_d$ is a Noetherian formal algebraic stack. Its underlying reduced substack $\mathcal{X}_{d,\operatorname{red}}$ is an algebraic stack of finite type over ${\mathbf F}_p$, and is equidimensional of dimension $[K:{\mathbf Q}_p] \binom{d}{2}$. The irreducible components of $\

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: EGmoduli
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Proposition 3.4: cegsB
  • Theorem 3.5
  • ...and 39 more