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THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 December 2014

TOBY GEE  and
MARK KISIN
TOBY GEE
Affiliation:
Department of Mathematics, Imperial College London SW7 2RH, UK; [email protected]
MARK KISIN
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA; [email protected]

Abstract

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We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$, $K$ a finite extension of $\mathbb{Q}_{p}$, for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 2 , 2014 , e1
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

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