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

Ana Caraiani, Matthew Emerton, Toby Gee and David Savitt

Vol. 19 (2025), No. 2, 287–312
DOI: 10.2140/ant.2025.19.287
Abstract

We establish a geometrization 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. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by means of a comparison of versal rings.

Keywords
moduli stacks, Galois representations, Breuil–Mezard conjecture
Mathematical Subject Classification
Primary: 11F80
Milestones
Received: 26 August 2022
Revised: 13 February 2024
Accepted: 29 April 2024
Published: 31 January 2025
Authors
Ana Caraiani
Department of Mathematics
Imperial College London
London
United Kingdom
Matthew Emerton
Department of Mathematics
University of Chicago
Chicago, IL
United States
Toby Gee
Mathematics Department
Imperial College London
London
United Kingdom
David Savitt
Department of Mathematics
Johns Hopkins University
Baltimore, MD
United States

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