23 problems

A local Shimura variety is associated with a local Shimura datum $(G,\mu,b)$, and its infinite-level space is obtained by passing to the inverse limit over compact open level subgr…

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 $(G,\{\mu\},b)$ 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…

Perfectoidness conjecture. The space $\mathcal{M}_{(G,\mu,b),\infty}$ is representable by a perfectoid space. This is known for local Shimura varieties of Abelian type; the paper g…

Let $(G,\mu,b)$ be an integral local Shimura datum. Denote by $\mathcal{M}^{\textup{int}}_{(G,\mu,b)}$ the integral model for the local Shimura variety at hyperspecial level. Repre…

Unconditional local Shimura conjecture. The natural transformation

Comparison-isomorphism conjecture. The morphism $\Upsilon^{\mathrm{sm}}$ is an isomorphism.

Bültel–Pappas conjecture. The functor

Let $(G,b,\mu)$ be a local Shimura datum and let ${\mathcal G}$ be a parahoric group scheme. Consider the local $v$-sheaf diagram … with the first map a ${\mathcal G}$-torsor and t…

Let $(G,b,\mu)$ be a local Shimura datum, let ${\mathcal G}$ be a parahoric group scheme, and let $x\in X_{\mathcal G}(b,\mu^{-1})(k)$. Let…

Let $(G,b,\mu)$ be a local Shimura datum, let ${\mathcal G}$ be a parahoric group scheme, and let $x\in X_{\mathcal G}(b,\mu^{-1})(k)$. Local tube representability conjecture. If…

Let $(G,b,\mu)$ be a local Shimura datum, let ${\mathcal G}$ be a parahoric group scheme, and let $k$ be the residue field. For a point $x\in X_{\mathcal G}(b,\mu^{-1})(k)$, write…

Integral-module conjecture. Under these hypotheses, there exists a locally free $\mathcal{O}^+$-module of finite rank on $\mathrm{Sh}_K(G,b,\mu)$ such that the conclusion of the mi…

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…

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 a reductive group and let $\mu$ be a minuscule cocharacter. Write $[b]$ for a basic $\sigma$-conjugacy class and let $(G,\{\mu\})$ be fully Hodge–Newton decomposable in…

Let $(G,[b],\{\mu\})$ be a local Shimura datum with local reflex field $E$, and let $\mathscr{F}\ell_{G,\mu}^{a}\subset\mathscr{F}\ell_{G,\mu}^{wa}$ be the open subspace expected t…

Assume that $G$ is quasi-split and let $(G,[b],\{\mu\})$ be a non-basic local Shimura datum. Let $L$ be a Levi subgroup, assume that $J_b$ is an inner form of a Levi subgroup conta…

Let $(G,[b],\{\mu\})$ and its dual $(G^\vee,[b^\vee],\{\mu^\vee\})$ be as in the duality setup. Let $\pi$ and $\rho$ be irreducible representations of $G(F)$ and $G^\vee(F)$, respe…

Let $(G,[b],\{\mu\})$ be a basic local Shimura datum, and let $(G^\vee,[b^\vee],\{\mu^\vee\})$ be its dual datum, with $G^\vee=J_b$, $[b^\vee]=[b^{-1}]$, and…

Assume that $G_{\mathrm{der}}$ is simply connected. Let $D=G_{\mathrm{ab}}=G/G_{\mathrm{der}}$ be the maximal abelian quotient of $G$, and let $\delta:G\rightarrow D$ be the projec…

Let $\mathcal{D}_{\mathbb{Z}_p}$ be an integral Rapoport–Zink datum such that $[b]\in B(G,\{\mu\})$, and suppose that the group scheme $\mathcal{G}$ is parahoric. Non-emptiness con…

Let $\mathcal{D}_{\mathbb{Z}_p}$ be an integral Rapoport–Zink datum such that $\mathcal{G}$ is a parahoric group scheme. The associated rigid space is equipped with an action of…