17 problems

Let $K^p$ be a neat compact open subgroup of $d53e({\mathbb A_f^p})$, let $K=K_pK^p$, and let $(\mathsf{G},\mathsf{X})$ be a Shimura datum satisfying…

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

Let $(G,X)$ be a Shimura datum, let $p$ be a prime, let $K_p$ be a hyperspecial subgroup, let $E$ be the reflex field, and let $v$ be a place of $E$ over $p$. An integral canonical…

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…

Let $E$ be the reflex field of the Shimura variety $\operatorname{Sh}=\operatorname{Sh}_K(G,h)$, let $\mathfrak p$ be a prime of $E$ above $p$, and let $\mathcal O_{\mathfrak p}$ b…

Very-good integral Hodge embedding conjecture. Any good integral Hodge embedding is also very good.

Let $G=\operatorname{SL}_2(\mathbb{F}_p)$ with $p\geq3$ prime. For each of the two irreducible Weil characters $\xi_1$ and $\xi_2$ arising from the reducible principal series, of d…

Let $\rho:G\rightarrow\operatorname{GL}(V)$ be an irreducible complex representation of a finite group $G$ of exponent $e$. A model for $\rho$ is a choice of basis of $V$ in which…

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…

Let $({\sf G},X)$ be an arbitrary Shimura datum and let ${\sf K}={\sf K}_p{\sf K}^p$, with ${\sf K}_p={\mathcal G}(\mathbb Z_p)$ for a connected parahoric model ${\mathcal G}$ of…

Choice-independence conjecture. The schemes ${\mathscr S}_K({\mathbb G},X)$ produced by these results are independent of the choices made in their construction.

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…