The Breuil-Mézard conjecture for non-scalar split residual representations (Q2788609)

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 \((k,\tau,\psi)\).NEWLINENEWLINE The proof consists of two parts: first following the strategy in [\textit{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 [\textit{M. Kisin}, J. Am. Math. Soc. 22, No. 3, 641--690 (2009; Zbl 1251.11045)].