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A geometric perspective on the Breuil-Mézard conjecture. (English) Zbl 1318.11061
“Let \(p>2\) 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 \(\mathbb{Q}_{p}\) [C. Breuil and A. Mézard, Duke Math. J. 115, No. 2, 205–310 (2002; Zbl 1042.11030)]. 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.”
The authors give in more detail their geometric refinement of the original Breuil-Mézard conjecture in subsetion 1.1.
“In an appendix we establish a technical result that allows us to realize representations of local Glois groups as restrictions of automorphic representations of global Galois groups.”

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
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