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The Breuil-Mézard conjecture for non-scalar split residual representations. (La conjecture de Breuil-Mézard pour les représentations résiduelles scindées non scalaires.) (English. French summary) Zbl 1334.11041
Let $$p>5$$ be a prime number, let $$E$$ be a sufficiently large finite extension of $$\mathbb{Q}_p$$. The authors give an essentially local proof of the cycle-theoretic version of the Breuil-Mézard conjecture on the equality between the Hilbert-Samuel multiplicity of a certain locus of deformations of a $$p$$-adic Hodge type $$(k,\tau,\psi)$$ with values in $$\mathrm{GL}_2(E)$$ and a certain sum of representation-theoretic multiplicities of $$\mathrm{GL}_2(\mathbb{Z}_p)$$ associated with $$(k,\tau,\psi)$$.
The proof consists of two parts: first following the strategy in [V. Paškūnas, Duke Math. J. 164, No. 2, 297–359 (2015; Zbl 1376.11049)], the authors prove the analogous statement for multiplicities of pseudo-deformation rings; then the conjecture is deduced by comparing multiplicities of the deformations rings $$R^{\mathrm{ps},\psi}$$, $$R_{\mathfrak{q}_i}^{\mathrm{peu},\psi}$$ and $$\widehat{R}_{\mathfrak{p}_i}^{\mathrm{ver},\psi}$$ over various prime ideals $$\mathfrak{p}_i$$ of $$R^{\mathrm{ver},\psi}$$. As a consequence, this allows the authors to remove the local restriction in the proof of the Fontaine-Mazur conjecture in [M. Kisin, J. Am. Math. Soc. 22, No. 3, 641–690 (2009; Zbl 1251.11045)].

##### MSC:
 11F80 Galois representations 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F85 $$p$$-adic theory, local fields 22E50 Representations of Lie and linear algebraic groups over local fields
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