66 problems

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 $K$ 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…

Let $\mathscr{F}\!\ell$ be the flag variety, let $N_0$ be the Fontaine operator, let $d$ be the relevant cohomological degree, and let $\mathfrak{m}$ be the specified Hecke ideal.…

Frobenius valuation conjecture. The valuations of the eigenvalues of $\Phi(\theta)$ are nonpositive integers; among them, listed with multiplicities, the number of times that each…

Gamma-class overconvergence conjecture. For the quantum connection on any monotone symplectic manifold, the Frobenius structure with constant term above, where $b=\Gamma_p(TM)$, is…

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…

Light profiniteness conjecture. Étale locally, the associated light condensed anima is a light profinite anima.

Let $\mathcal{S}h$ be a Shimura variety and let $Z\subseteq\mathcal{S}h$ be a positive-dimensional subvariety. Denote by the Hecke orbit of $Z$ the union of its images under the He…

Let $X/C$ be a smooth proper connected variety of pure dimension $d$ with globally generated $1$-forms, and fix a point $x\in X(C)$. Let … be a closed subgroup. The canonical map s…

Weight-monodromy conjecture. The two filtrations agree up to the shift specified by