Geometric Breuil–Mézard conjecture for two-dimensional representations

Let ${\bar{r}}:G_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb F)$ and the deformation rings and multiplicities $a_{m,n}$ and $a^{\operatorname{cr}}_{m,n}$ be as above. For each $0\leq m\leq p-2$ and $0\leq n\leq p-1$, a cycle records the corresponding Serre weight. Geometric Breuil–Mézard conjecture. For each $0\leq m\leq p-2$ and $0\leq n\leq p-1$, there is a cycle $\mathcal C_{m,n}$ depending only on $m$, $n$, and ${\bar{r}}$, such that for any $a$, $b$, and $\tau$ with $\det\tau=\varepsilon^{-2a-b-1}\psi|_{I_{\mathbb Q_p}}$,

$$Z(R^{\square,\psi}(a,b,\tau,{\bar{r}})/\pi)=\sum_{m,n}a_{m,n}\mathcal C_{m,n},$$

and

$$Z(R_{\operatorname{cr}}^{\square,\psi}(a,b,\tau,{\bar{r}})/\pi)=\sum_{m,n}a^{\operatorname{cr}}_{m,n}\mathcal C_{m,n}.$$

This refines the multiplicity formula to an equality of cycles and is the geometric formulation of the local Breuil–Mézard conjecture.