19 problems

Let $k$ be a perfect field of characteristic $p>0$, let $W(k)$ be its ring of Witt vectors, and set $K=W(k)[1/p]$. Let $f:X\to S$ be a smooth proper morphism of $k$-schemes. For a…

Weak admissibility conjecture. The specialization map $\iota_{\widehat{F}}$ is an isomorphism, and

Let $X_\eta$ be the generic fibre of the strictly semi-stable scheme considered in the paper, let $\mathcal{E}^{\dagger}_K$ be the bounded Robba-ring coefficient field, and write…

Let $k$ be a perfect field of characteristic $p>0$, let $W(k)$ be its ring of Witt vectors, and let $X/k$ be a smooth variety. The overconvergent de Rham–Witt complex…

Berthelot–Tsuzuki conjecture. Under a smooth proper morphism of varieties over $k$, the higher direct image of a (over-)convergent ($F$-)isocrystal is still a (over-)convergent (…

Let $K$ be a complete valued field with residue field $k$, and suppose that $K$ satisfies the technical hypotheses in Assumption. Let $\Lambda$ be the coefficient ring, and let…

Let $k=\mathbb F_q$ be a finite field with $q=p^f$, let $L$ be the function field under consideration, let $X$ be a smooth projective variety over $L$, let $r$ be a codimension, an…

Let $R$ be a complete discrete valuation ring with fraction field $K$ and finite residue field $k$ of characteristic $p$, and let $X$ be a proper flat scheme over…

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

Let $F$) be a local field of residue characteristic $p>0$, let $X$ be a smooth and proper variety over $F$, and let $\ell$ be a prime different from $p$. Suppose that $X$ has semis…

Let $X\to S$ be proper, flat, generically smooth and regular. Let $\operatorname{sp}'$ be the specialization map from rigid cohomology to crystalline cohomology, and let $W_{i-1}$…

Let $X\to S$ be proper, flat and generically smooth, with special fibre $X_s$ over $k$ and geometric generic fibre $X_{\bar\eta}$. Let $H^i_{\operatorname{rig}}(X_s/k)$ be rigid co…

Let $X\to S\subseteq \mathbb A^1_{\mathbb F_q}\setminus\{0\}$ be a morphism of varieties over $\mathbb F_q$ with a good lift around $t=0$. Let…

A degeneration of projective varieties over a finite field of characteristic $p$ satisfying the paper's $p$-adic lifting assumption has an associated limiting Frobenius structure.…

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 $(S,S,\mathcal S)$ 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…