A geometric perspective on the Breuil–Mézard conjecture
By M. Emerton, T. Gee. Journal of the Institute of Mathematics of Jussieu, 2011
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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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/ac5148d568f25524653dbc870b1e640d2c40b27c
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