11 problems

Let $k(v)$ be the residue field at $v$, let $\kappa$ be a perfect field over $k(v)$, and let $x\in \mathscr{S}_K(\kappa)$. For each prime $\ell$, write $I_{\ell,x}$ for the corresp…

Integral-model conjecture. The composition

Let $(G,\{\mu\},b)$ be a local Shimura datum, let ${\mathcal G}$ be parahoric, set $K_0={\mathcal G}(\mathbb Z_p)$, and let $K_1$ be the kernel of the reduction map to the reductiv…

Assume the divisor conjecture and let $pi:{rm Sh}{K1}(G,X)E$ be the generic-fiber $T{\mathcal G}(\mathbb Fp)$-torsor,…

Assume that $({\mathcal G},\{\mu\})$ is strictly convex, that the divisor conjecture holds, and that an $O_E$-integral model ${\mathcal S}_{K_0}$ of ${\rm Sh}_{K_0}(G,X)_E$ with it…

Canonical integral models conjecture. For every such Shimura datum and quasi-parahoric model, there exists a system of canonical integral models…

Let $(\mathcal{G},\mathbf{G},\mathbf{X})$ be a parahoric Shimura datum, and let $\{\operatorname{Sh}_{\sf K}(\mathbf{G},\mathbf{X})\}_{{\sf K}^p}$ denote the associated system of S…

Let ${\mathscr S}_{\sf K}$ be an integral model of a Shimura variety equipped with its $v$-sheaf theoretic local model diagram. A scheme-theoretic local model diagram is a diagram…

The integral-model conjecture. There is a scheme ${\mathscr S}_{K_p}({\mathbb G},X)$ over … with a right action of ${\mathbb G}({\mathbb A}_f^p)$ satisfying: (a) sufficiently small…

Let $(G,\mathcal{X})$ be the Shimura datum under consideration, let $G^{\operatorname{ad}}$ be its adjoint group, let $N\in\mathbb{N}$ with $N\geq 3$, and let $\mathcal{N}(N)$ be t…

Let $(G_1,\mathcal{X}_1)$ be a Shimura pair such that $G_{1,\mathbb{Q}_p}$ is unramified. Let $H_1$ be a hyperspecial subgroup of $G_1(\mathbb{Q}_p)$, and let $v_1$ be a prime of…