12 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…

Let convergent and overconvergent $F$-isocrystals be coefficient objects in the relevant $p$-adic cohomological categories, with morphisms considered in the convergent category. Ts…

Let $k$ be a perfect field, let $X$ be a smooth $k$-variety, and let $\mathcal{E}$ be an overconvergent $F^a$-isocrystal on $X$, where $F^a$ denotes a Frobenius structure of order…

Let $G$ be a connected reductive algebraic group over $\mathbb{Q}_p$, and let ${\mathcal F}$ be the homogeneous space of filtrations on an $F$-isocrystal with $G$-structure. Write…

Le Stum's conjecture. For any pair $(X,Y)$ over $k$, the functor $\mathrm{sp}_!$ is an equivalence of categories.

Let $C_0$ be a smooth projective curve over the finite field $\mathbb F_q$ of characteristic $p>0$, let $C$ be its base change to $\overline{\mathbb F}_q$, and let $M_K$ and…

Let $X$ be a simply connected variety over a field of characteristic $p>0$, and let an $F$-isocrystal mean an isocrystal on $X$ equipped with Frobenius structure. The F-isocrystal…

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 $Y$ be the variety appearing in the definition of companions. For an abstract field isomorphism $\tau$ between the relevant algebraic closures of coefficient fields, a $\tau$-c…

Let $X$ be a variety, let $\operatorname{\mathbf{F-Isoc}}^\dagger(X)$ denote the category of overconvergent $F$-isocrystals on $X$, and let $C \subseteq X$ be a curve. Curve restri…

Let $X$ be a variety, let $Y$ be a boundary or compactification datum, and let $C$ denote a curve contained in the indicated variety. Write $\operatorname{\mathbf{F-Isoc}}(X)$ and…

Let $X$ be the fixed variety, and let $\operatorname{\mathbf{F-Isoc}}(X)$ and $\operatorname{\mathbf{F-Isoc}}^\dagger(X)$ denote the categories of convergent and overconvergent $F$…