17 problems

Let $k$ be the base field, let $Y$ be a smooth $k$-variety, let $E$ be an overconvergent $F$-isocrystal on $Y$, and let $\operatorname{sp}_{Y+}(E)$ denote its specialization as an…

Let $\mathcal{V}$ be a complete discrete valuation ring of mixed characteristics $(0,p)$ with perfect residue field $k$. Let $\mathfrak{X}$ be a separated smooth formal…

Let $Y$ be a smooth variety over $k$ (that is, a separated smooth $k$-scheme of finite type), and let $E$ be an overconvergent $F$-isocrystal on $Y$. A morphism…

Let $Y_0$ be a $k$-scheme, let $X_0$ be a smooth ambient scheme when specified, and let $\rho_{Y_0,X_0}:\mathrm{Isoc}^{\dag}(Y_0/K)\to\mathrm{Isoc}^{\dag}(Y_0,X_0/K)$ and…

Let $k$ be a field of characteristic $p>0$, let $K$ be the fraction field of a complete mixed characteristic discrete valuation ring with residue field $k$, and let $X$ be a smooth…

Let $I$ be an index set, let $X_i$ be the varieties equipped with partial Frobenius actions, and let $X/\Phi$ be the associated partial Frobenius stack. Write…

Crystalline analogue conjecture. The crystalline category analogous to the category of Weil $\overline{\mathbb{Q}}_\ell$-sheaves is the category of overconvergent $F$-isocrystals.

Stronger Shiho conjecture. There exists a unique overconvergent $(F)$-isocrystal $\widetilde E$ on $(S,\overline{S})$ such that

Shiho's conjecture. There exists a unique overconvergent $(F)$-isocrystal $\widetilde E$ on $(S,\overline{S})$ whose restriction to…

Tsuzuki's conjecture. There is a unique object

Berthelot's conjecture. There is a unique object

Equivalence conjecture. There is a canonical equivalence of categories

Canonical equivalence conjecture. The canonical functor

Let $X$ be a variety over a perfect field of characteristic $p>0$, and let $\mathcal{E}$ be an overconvergent $F$-isocrystal on $X$. Semistable reduction conjecture. The isocrystal…

Let $S:=\operatorname{Spec} k$, let $\mathcal{S}:=\operatorname{Spf} V$, and consider separated finite-type pairs and triples over $(S,S)$ and $(S,S,\mathcal{S})$. Let…

Let $k$ be the perfect field and $\mathcal{S}$ the fixed formal base. Let $X$ be a smooth separated $k$-scheme of finite type, and let $X\hookrightarrow\overline{X}$ be an open imm…

Let $(S,S,\mathcal S)$ be a triple over $(B,B,\mathcal B)$, and suppose we are given a diagram of pairs … such that $f:\overline{X}\to\overline{Y}$ is proper, $f^{-1}(Y)=X$, and…