4 problems

Let $G=\operatorname{GL}_n$. For each Serre weight $\sigma$ of $G(\mathbf{F}_p)$, let $\mathcal{Z}(\sigma)$ be a cycle in the $d$-dimensional cycle group of the reduced Emerton–Gee…

Let $\sigma$ range over parahoric Serre weights, and let $\mathcal{X}^{\operatorname{crys},\lambda,\tau}_{{^{L}\!G},\mathbb{F}_p}$ have cycle class in the Chow group of…

Let $\lambda$ be a Hodge type, viewed as an element of the character lattice of $G$, and let $\tau$ be a tame inertial type. Let $V(\lambda-\eta_{\mathbb{Q}_p})$ denote the restric…

Let ${\mathcal X}_{{\mathrm{Sym}},K,\mathrm{red}}$ be the reduced Emerton--Gee stack, and let ${\mathcal Z}_{\lambda,\tau}$ be the $4f$-dimensional cycle attached to a regular Hodg…