Breuil–Mézard conjecture for two-dimensional Galois representations

Let $L/\mathbb{Q}_p$ be a finite extension, let $E$ be a coefficient field, let $W_L$ be the set of Serre weights, and let $\overline{\rho}_L:G_L\to\mathrm{GL}_2(\mathbb{F})$ be a continuous representation. For a weight $\lambda\in(\mathbb{Z}_+^2)^{\operatorname{Hom}_{\mathbb{Q}_p}(L,E)}$ and an inertia type $\tau$, let $R^{\mathrm{crys},\lambda,\tau}_{\overline{\rho}_L}$ and $R^{\mathrm{ss},\lambda,\tau}_{\overline{\rho}_L}$ be the corresponding crystalline and semistable deformation rings, let $n_k^{\mathrm{crys}}(\lambda,\tau)$ and $n_k^{\mathrm{ss}}(\lambda,\tau)$ be the specified multiplicities, and let $\mu_k(\overline{\rho}_L)$ be the Breuil–Mézard multiplicities. Breuil–Mézard conjecture. For every such $\lambda$ and $\tau$,

$$e(R^{\mathrm{crys},\lambda,\tau}_{\overline{\rho}_L}/\varpi)=\sum_{k\in W_L}n_k^{\mathrm{crys}}(\lambda,\tau)\mu_k(\overline{\rho}_L),$$

and

$$e(R^{\mathrm{ss},\lambda,\tau}_{\overline{\rho}_L}/\varpi)=\sum_{k\in W_L}n_k^{\mathrm{ss}}(\lambda,\tau)\mu_k(\overline{\rho}_L).$$

The conjecture predicts Hilbert–Samuel multiplicities of deformation rings from Serre-weight multiplicities. The supplied text recalls results identifying the crystalline weight support and cites established theorems for parts of the framework, but does not resolve the displayed general formulas.