The versal Breuil–Mézard conjecture

Let $K/\mathbb{Q}_p$ be a finite extension, let $\overline{\rho}:G_K\rightarrow\operatorname{GL}_n(\mathbb{F})$ be continuous, and regard $\overline{\rho}$ as an $\mathbb{F}$-point of the Emerton–Gee stack $\mathcal{X}_n$. Fix a versal ring $R_{\overline{\rho}}^{\mathrm{ver}}$ and let $R_{\overline{\rho}}^{\mathrm{alg}}$ be the induced ring for the reduced stack. For an extremal type $(\lambda+\eta,\tau)$, let $\mathcal{Z}_{\lambda,\tau}(\overline{\rho})$ be the pullback cycle in $\operatorname{Spec}R_{\overline{\rho}}^{\mathrm{alg}}$. Versal Breuil–Mézard conjecture. For every set $\mathcal{S}$ of extremal types and every $\sigma\in\mathrm{JH}(\overline{\sigma}(\mathcal{S}))$, there exist effective cycles $\mathcal{Z}_\sigma(\overline{\rho})$ in $\operatorname{Spec}R_{\overline{\rho}}^{\mathrm{alg}}$ such that, for all $(\lambda+\eta,\tau)\in\mathcal{S}$,

$$\mathcal{Z}_{\lambda,\tau}(\overline{\rho})=\sum_\sigma[\overline{\sigma}(\lambda,\tau):\sigma]\mathcal{Z}_\sigma(\overline{\rho}).$$

This is the local, versal-ring realization of the geometric Breuil–Mézard formula and relates stack-theoretic cycles to deformation rings at individual residual representations.