Breuil–Mézard conjecture for tame determinant types

Fix an odd prime $p$, a finite extension $E$ of ${\mathbb Q}_p$ with residue field ${\mathbf k}_E$, and a potentially crystalline representation $\rho:G_{{\mathbb Q}_p}\rightarrow {\rm GL}_2(E)$ with Galois type $\tau$. Let $2\leq k<p$, let $\overline{\rho}:G_{{\mathbb Q}_p}\rightarrow {\rm GL}_2({\mathbf k}_E)$ have trivial endomorphisms, and let $R(k,\tau,\overline{\rho})_{\mathcal O_E}$ be the associated potentially semistable deformation ring for Hodge–Tate weights $(0,k-1)$ and fixed determinant. Write $\mu_{\mathrm{gal}}(k,\tau,\overline{\rho})$ for its special-fibre Samuel multiplicity, and let $\mu_{\mathrm{aut}}(k,\tau,\overline{\rho})$ be the corresponding automorphic integer. Breuil–Mézard conjecture. If $\det(\tau)$ is tame, then

$$\mu_{\mathrm{gal}}(k,\tau,\overline{\rho})=\mu_{\mathrm{aut}}(k,\tau,\overline{\rho}).$$

This is the multiplicity formula relating the geometry of local Galois deformation rings to automorphic data. The source presents it as the first part of the Breuil–Mézard conjectures and proves the relevant conjectural statements in the paper's potentially crystalline setting.