70 problems

For every nonsingular Hermitian variety $X$ of dimension $2m$ over a finite field, let $h$ denote the hyperplane class and let $N^m(X)_{\mathbb{R}}$ be the space of codimension-$m$…

For every number field $F$, prime $p$, Hilbert modular variety $X/F$, and cohomological degree $i$, the locally analytic $p$-adically completed cohomology $\widetilde{H}^{i}(X)$ sh…

For a rigid analytic space $X$ over $\mathbb{C}_p$ and a profinite pro-étale torsor $Y\to X$, if $Y$ is affinoid perfectoid, then the associated geometric Sen morphism is surjectiv…

Let $K$ be a number field, let $S$ be a finite set of finite places of $K$, and let $X=\mathbb{P}^{1}\smallsetminus\{0,1,\infty\}$. Kim's conjecture asserts that the motivic Chabau…

Let $X$ be a proper smooth variety over a local field $k$, and let $V$ be its $i$th $\ell$-adic étale cohomology group. Write $N$ for the monodromy operator, let $W_\bullet$ be the…

Let $D=0+1+\lambda+\infty$ on $\mathbb{P}^1$, and consider logarithmic graded semistable Higgs bundles on $(\mathbb{P}^1,D)$ whose underlying graded vector bundle is isomorphic to…

Let $X$ be a smooth projective variety over $\mathbb{C}_p$, and let $X^{\mathrm{an}}$ denote its analytification. Kodaira-dimension conjecture. If there is a $p$-adic analytic map…

Let $p\geq 5$, and let $\overline{\rho}$ be a residual Galois representation that is regular in the sense specified in the paper. For each…

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 $K$ be a $p$-adically closed field and let $A$ be an infinite definable subset of $K^n$ that is open in its closure. Let $(B_k)_{k\leq q}$ be a finite collection of closed defi…

Let $V$ be a smooth projective geometrically integral variety over a $p$-adic field $k$. Let $A_0(V)$ be the degree-zero subgroup of the Chow group of zero-cycles, and let … be the…

Let $Sh_{K_p}$ be a PEL Shimura variety with standard parahoric level subgroup $K_p$, and let … be the map sending a point to the $c3$-conjugacy class of its associated isocrystal.…

Let ${\mathbb C}_p$ be the completion of the algebraic closure of ${\mathbb Q}_p$, with $v$ the extension of the $p$-adic valuation, and let … be its ring of integers. Let…

Stable supersingular-component conjecture. There is a semistable covering of $X_0(p^3)$ defined over $R_p$ such that, for each such $A$, one connected component of the supersingula…

CM residue-disk placement conjecture. The $j$-invariants of all such curves $E$ lie in $2(p+1)/i$ residue disks inside the corresponding Atkin–Lehner circle, with $(p+1)/i$ disks f…

CM placement conjecture at 13. If

CM placement conjecture at 7. If

Let $\tau,\omega\in\mathcal{H}_p$ be RM points with $\omega$ not in the $G$-orbit of $\tau$, let $\mathrm{T}\in\mathcal{I}_p$ be a Hecke operator, and let…

Let $X$ be a smooth projective rigid analytic space over $\mathbb{C}_p$, and let $G$ be a reductive group defined over $\mathbb{C}_p$. A $G(\mathbb{C}_p)$-representation of…

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 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 $(G,X)$ be a Shimura datum with associated $G(\mathbb{C})$-conjugacy class of minuscule cocharacters $[\mu^{-1}]$ and reflex field $E_0$. Fix a prime $p$, let $E$ be the comple…

Let $G$ be the group in the source, let $\mu$ be the specified minuscule cocharacter, let $GDR^{\Psi-\mathrm{la}}_{\lambda}$ be the geometric locally analytic de Rham complex of we…

Let $G_{\mathbb{Q}_p}:=\operatorname{Res}_{F_w/\mathbb{Q}_p}\mathrm{GL}_{2,F_w}$, let $\mu$ be the specified minuscule cocharacter, and let $\mathscr{F}\!\ell^b$ be the Newton stra…

Let $\mathscr{F}\!\ell$ be the flag variety, let $GDR^{\mathrm{la}}_0$ be the geometric locally analytic de Rham complex, let $d$ be the relevant cohomological degree, and let…