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A geometric perspective on the Breuil–Mézard conjecture

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 June 2013

Matthew Emerton  and
Toby Gee
Matthew Emerton
Affiliation:
Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208, USA (emerton@math.northwestern.edu; gee@math.northwestern.edu)
Toby Gee
Affiliation:
Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208, USA (emerton@math.northwestern.edu; gee@math.northwestern.edu)
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Abstract

Let $p\gt 2$ be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod $p$ representations of the absolute Galois group of ${ \mathbb{Q} }_{p} $. We also state a conjectural generalization to $n$-dimensional representations of the absolute Galois group of an arbitrary finite extension of ${ \mathbb{Q} }_{p} $, and give a conditional proof of this conjecture, subject to a certain $R= \mathbb{T} $-type theorem together with a strong version of the weight part of Serre’s conjecture for rank $n$ unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate representations.

Keywords

automorphic formsGalois representations

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 1 , January 2014 , pp. 183 - 223
Copyright
©Cambridge University Press 2013 

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