25 problems

Corrected local model flatness conjecture. The scheme $M$ is flat over

Pappas's spin flatness conjecture. The scheme $M_I$ is flat over $\mathcal O_E$; equivalently,

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…

Reducedness conjecture. The scheme $\overline N$ is reduced. This reducedness assertion is used to identify the special fiber of the canonical flat model and to establish its norma…

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 $F/F_0$ 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 $F$ 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 $X_L$ 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 $\underline{\mathcal{G}}$ 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 $\calM_{\calG,\mu}^{\on{sch}}$ be the scheme-theoretic local model attached to a parahoric group scheme $\calG$ and a minuscule cocharacter $\mu$. Pseudo-rationality conjecture…

Full p-adic coherence conjecture. This natural morphism is always an isomorphism. The assertion is already known under some hypotheses, while the paper identifies a small number of…

Local model conjecture. For every closed point $x\in |({X_{\mathscr{G}}^{\leq\mu}(b)})_{{k_F}}|$ there exists a closed point $y\in |({{\mathcal{A}}_{\mathscr{G},\mu}})_{k_F}|$ such…

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 $(G,\{\mu\},K)$ be a local model triple, with $G$ split over a tame extension, and let ${\mathbb M}^{\mathrm{loc}}_K(G,\{\mu\})$ be the associated flat projective local model o…

The local model conjecture. There exists a projective and flat scheme … over … which supports an action of ${\mathcal G}\otimes_{\mathbb Z_p}{\mathcal O}_E$ and is such that: (i) t…

Test function conjecture. As elements of $\mathcal Z(G(E_0),\mathcal G(\mathcal O_{E_0}))$,

Let $(r,s)$ be a signature and let $I\subset\{0,\dotsc,m\}$ be nonempty. Let $M_I^{\mathrm{naive}}$ be the naive local model, $M_I$ the closed subscheme cut out by the wedge and st…

Let $F/F_0$ be a ramified quadratic extension of discretely valued non-Archimedean fields with common residue field of characteristic not $2$, let $n\geq 2$, let $(r,s)$ be any sig…

Let $Sh_{K_p}$ be the Shimura variety, let $\mathcal M_{K_p}/\mathcal O_E$ be a natural integral model, and let $R\Psi_x$ denote the nearby-cycles complex at a point $x$ of its spe…

Pappas–Rapoport flatness conjecture. The scheme $P$ is flat over

Görtz's flatness conjecture. For any self-dual periodic ${\mathcal O}_F$-lattice chain ${\mathcal L}$, $M^{\mathrm{naive}}_{G,\{\mu\},{\mathcal L}}$ is flat over

Pappas–Rapoport spin-condition conjecture. For any periodic self-dual lattice chain ${\mathcal L}$,