31 problems

Let $(\mathrm{G},b,\mu)$ be a PEL-type Rapoport–Zink datum, with $\mathrm{G}$ a connected reductive group over $\mathbb{Q}_p$, $\mu$ a minuscule cocharacter, and…

Let $L_0=\mathbb Z_p^n$ and $L_0^\vee=p^{-1}\mathbb Z_p^t\oplus\mathbb Z_p^{n-t}$, and let $\mathrm{GL}(L_0,L_0^\vee)$ be the corresponding parahoric subgroup. For a regular semisi…

Arithmetic transfer conjecture for odd type $(r,0)$. There exists $\varphi'\in C_c^\infty(G')$ transferring to $(c_r^2\\,\mathbf{1}_{K_n^{[r]}\times K_{n+1}},0)$ such that, for eve…

Let $G$ be a connected, quasisplit reductive group over $4p$, let $4mu$ be a dominant cocharacter, and let $b\in \mathbf{B}(G,\mu)$. Let $M_b\subset M_S\subset G$ be standard ratio…

Let $v \in {}^{I}\operatorname{Adm}(\{\mu'\})_{0}$, and write $v=v_Ju_{\mathbf a}$ as in Proposition; namely, for some $J\in\mathcal P_{0,I}$ and…

Harris–Viehmann conjecture.

Cohomological smoothness conjecture. The structure morphism

Cohomological smoothness conjecture. The structure morphism

Let $0\leq r\leq n+1$, let $\varepsilon$ be the parity of $r$, and let $W_\varepsilon$ and $W_{\varepsilon+1}$ be the hermitian spaces of dimension $n+1$ with opposite invariants.…

For $0\leq r\leq n$, let $\widetilde{\mathcal{N}}^{[r]}_n$ be the formal scheme constructed from the relevant Rapoport--Zink spaces, and let $p$ be the residue characteristic. Regu…

Hamacher–Kim uniformization conjecture. For $\ell=\overline{\mathbb{F}}_p$, there should be a uniformization map

Let $\mathcal{N}^{[h]}_n$ be the Rapoport–Zink space and let $L^{\flat}\subset \mathbb{V}$ be an $O_F$-lattice of rank $n-1$. Let $\mathcal{Z}(x)$ and $\mathcal{Y}(x)$ denote the c…

Let $\mathcal{N}_{s,n}$ be the moduli space of quadruples defining the Rapoport–Zink space for $\operatorname{GU}(s,n-s)$ over a ramified prime, with $n\geq 3$ and $s\leq n$, and s…

Let $t\neq t'$ and let $A$ be any closed formal subscheme of the Rapoport–Zink space $\mathcal{N}_n$. The correspondence $\mathcal{T}_n^{\leq t}$ is obtained by composing the corre…

Let $(G,b,\mu)$ be the local Shimura data appearing in the paper, let $\mathcal{H}^{\mathfrak{e}}$ be an endoscopic datum, and let $\operatorname{Mant}_{G,b,\mu}$ and…

Let $\mathit{k}$ be the residue field. Consider a Rapoport–Zink space and the corresponding affine Deligne–Lusztig set, with their natural geometric structures; write the perfectio…

Let $V$ be the relevant Hermitian space and let $x_1,\ldots,x_n\in V$ be a $\bm{k}$-basis. On a fixed connected component $M^\circ$ of the formal $\breve{\mathcal O}_{\bm{k}}$-sche…

Local Rapoport–Zink conjecture. There exists a formal scheme ${\mathfrak X}_K(\{\mu\},b)$ over $\operatorname{Spf}({\mathcal O}_E)$ with $J_b(\mathbb Q_p)$-action, locally formally…

Let $(G,\mu)$ be a local Shimura datum as in the source, let $b\in\mathbf{B}(G,\mu)$, and let $M_S\subset M_b$ be a standard Levi subgroup. Let $\rho$ be a supercuspidal representa…

Let $(G,b,\mu)$ be the datum of an unramified Rapoport–Zink space of EL-type. Let $M_b=M_1\times\cdots\times M_k$ be the natural product decomposition of the standard Levi subgroup…

Fix a local unramified Shimura datum $(G,[b],\{\mu\})$, choose an integral representative $\mu$ and a representative $b$ such that $b=u\mu^\sigma(p)$, and let…

Let $G$ be the group in the local unramified Shimura datum, let $b$ be a representative of its $\sigma$-conjugacy class, and let $W$ be the Witt-vector ring used in the constructio…

Let $(G,\{\mu\},[b])$ be an unramified local Shimura datum, with $G$ a connected reductive algebraic group over $\mathbb Q_p$, $\{\mu\}$ a conjugacy class of minuscule cocharacters…

Let $G$ be the local reductive group related to a Rapoport-Zink space, and let $(\mathcal{M}_K)_K$ be its tower with level $K$. Kottwitz's conjecture. The $\ell$-adic cohomology of…

Let $S(E_0)$ be the relevant symmetric space, let $U(J_1)$ be the unitary group attached to the odd hermitian form, and let $Ω(γ)$, $\partial O_γ(1_{S(\mathcal{O}_{E_0})})$, and…