54 problems

Let $F/F_0$ be a quadratic extension, let $(V,\phi)$ be a symmetric space of dimension $2n+2$, and let $I\subset [0,n]$ be non-empty. Let $\RM^\pm_I$ be the spin local model over…

Let $r+s=n$, with signature $(r,s)$, and let $M_I^{\mathrm{loc}}(r,s)$ be the scheme-theoretic closure of the generic fiber of the naive local model $M_I^{\mathrm{naive}}(r,s)$. Le…

Let $S=S(G,h)$ be a Shimura variety with $G\otimes_{\mathbb{Q}}\mathbb{R}=GU(r,s)$, and let $M_{r,s}$ be the associated local model over $\operatorname{Spec}(R)$, where $R$ is the…

Let $\underline{\mathcal{G}}$ be the relevant parahoric group scheme and let $\mu$ be the specified cocharacter datum, and write $\widetilde{M}_{\underline{\mathcal{G}},\mu}$ for t…

Let $G$ be a reductive group with parahoric group scheme $\mathcal{G}$, and let $\mu$ be a geometric conjugacy class of cocharacters, with the auxiliary data required in mixed char…

Vertex-wise admissibility conjecture. Admissibility has a vertex-by-vertex characterization; equivalently,

Coherence conjecture. Provided that $\mu$ is a sum of minuscule coweights for $G_{\rm ad}$,

Let $\mathbf M^{\mathrm{naive}}_{GSp_{2g}}$ denote the naive local model for the symplectic similitude group $GSp_{2g}$ in the ramified setting considered in the paper. Flatness co…

The semistable-reduction conjecture. There exists a ${\cal G}\otimes_{{\cal O}_F}{\cal O}_E$-equivariant blowing up in the special fiber

Let $M_I^{\rm can}$ be the canonical local model, defined as the scheme-theoretic image of the morphism from the splitting model $\mathcal M_I$ to the naive local model…

Let $K/E$ be the extension introduced in the construction, let ${\cal M}_{\bf s}$ be the multiplicity space occurring in the decomposition of the nearby-cycle complex, and let…

Let $M^{\rm naive}$ be the naive local model of EL-type over ${\rm Spec}\,{{\cal O}}_E$, and let $M^{\rm loc}$ be the closed subscheme obtained by imposing the corresponding local-…

PR24's schematic local model conjecture. There exists a smooth morphism

Let $F/F_0$ be a quadratic extension, let $(V,\phi)$ be a symmetric space of dimension $2n+2$, and let $I\subset [0,n]$ be non-empty. Let $\RM^\naive_I$ be the naive local model ov…

Let ${\rm Adm}^{\mathcal G}(\{\mu\})$ be the admissible set and let $|\Delta|$ be the divisor map from this poset to the face poset ${\mathscr F}(\sigma_{{\mathcal G},\mu})$. For…

Let $(G,\{\mu\},{\mathcal G})$ be a local model triple over $\mathbb{Q}_p$. Suppose that $G$ splits over a tamely ramified extension of $\mathbb{Q}_p$ and that $p$ does not divide…

Let $F$ be a discretely valued field with ring of integers $\CO$, let $k$ be its residue field, and let $n\geq 1$. For a non-empty subset $I\subset[0,n]$, let $\Lambda_I$ be the as…

Let $F$ be a discretely valued field with ring of integers $\CO$, and let $G$ be the split even orthogonal group of rank $2n$. Let $\CL$ be a self-dual lattice chain, and let…

Let $F$ be a complete discretely valued field with ring of integers $\mathcal{O}$, uniformizer $\pi$, and residue field $k$ of characteristic $p>2$. Let $V=F^{2n}$ with a split non…

Let $G$ be a reductive group with Shimura datum $(G,X)$, let $K=K_pK^p$ be a sufficiently small level, and let $\mathbb{M}_{\mathcal{G},\mu}$ be the normal scheme representing the…

Let $F/F_0$ be the ramified quadratic extension, let $V$ be a hermitian space over $F$ of signature $(r,s)$, and let $I$ be a nonempty subset of $\{0,\ldots,\lfloor n/2\rfloor\}$ s…

Let $M_I^{\mathrm{loc}}(r,s)$ be the local model and let $M_I(r,s)$ be the closed subscheme defined by the strengthened spin condition. Smithling's strengthened-spin conjecture. On…

Let $X_L$ be the local model with closed subschemes $X_w$, let $\widehat{\mathcal X}^{\flat}_{L,\widehat y}$ and $\widehat{\mathcal X}^{\flat}_{L,w,\widehat y}$ be the correspondin…

Let ${\mathcal U}$ and $\mathcal M$ denote the schemes defined in the paper, and consider dimensions $n=5,6,7,8$. Normality conjecture. The scheme ${\mathcal U}$, and hence…

Let $O_E$ be the reflex field's valuation ring, and consider the perfectoid local models introduced by Scholze and Weinstein as v-sheaves. Scholze–Weinstein's representability conj…