19 problems

Let $X_K$ be a smooth scheme of finite type over $K$. Let $H^{n}_{pst}(X_K)$ be the finite-dimensional $\mathcal{H}^{ur}_1$-module equipped with its increasing weight filtration…

Let $X$ be a regular $p$-adic formal scheme, and let $Y_\eta\to X_\eta$ be a proper smooth rigid space with analytically good reduction over $X_\eta$. For each $i\in\mathbb{N}$, an…

Let $X/\mathcal{O}_K$ be the fixed scheme, let $\mathfrak{S}$ be the Breuil–Kisin ring, and let $E$ be its Eisenstein polynomial. For the Breuil–Kisin cohomology module … write…

Main conjecture. For all $i\geqslant 0$,

Let $\mathcal{X}$ be a smooth proper $W(k)$-scheme, where $k$ is a perfect field of characteristic $p>0$. Suppose that $Z_0$ is a codimension $n$ cycle in…

Non-commutative crystalline comparison conjecture. There is an isomorphism of $B_{\operatorname{crys}}$-modules

Let $S$ be the base scheme, let $X$ be the relevant smooth variety over $S$ with boundary divisors $D_i$, and let $T_i$ prescribe the monodromy at infinity. Suppose there is one, o…

Let $K$ be a complete discretely valued $p$-adic field of characteristic $0$ with ring of integers $\mathcal{O}_K$ and perfect residue field $k$, and let $X$ be a smooth, proper sc…

Let $X$ be a smooth projective variety over the finite field $k=\operatorname{\mathbb{F}}_q$ of characteristic $p$, with $q=p^m$. Let $W(k)$ be the ring of $p$-typical Witt vectors…

Let $f\colon V\hookrightarrow U$ be an open sub-curve, let $u\in V({\mathbb{F}}_{q^e})$ be a base point, and let ${\mathcal{F}}$ be a convergent $F$-isocrystal on $U$. Write…

Let ${\mathcal{F}}$ be a convergent $F$-isocrystal on the curve $U$, fix $u\in U({\mathbb{F}}_{q^e})$, and let $\operatorname{Gr}({\mathcal{F}}/U,u)$ be its monodromy group over…

Let $U$ be the curve, let $K$ be the specified finite totally ramified extension, and let ${\mathcal{F}}$ be a convergent $F$-isocrystal on $U$. Fix $u\in U({\mathbb{F}}_{q^e})$, a…

Companion conjecture. Any algebraic $\ell$-adic coefficient object on $X$ admits an $\ell'$-adic companion.

Let $X$ be a smooth toric variety of dimension $n$, let $Y$ be a smooth hypersurface, and consider logarithmic crystalline cohomology $H^n_{\operatorname{cris}}(X_0,Y_0)$ together…

Let $f$ define a nonsingular hypersurface $X_f=\{f=0\}$, and let $H_f=H^{\cdot}_{dR}(X_f)$ be its de Rham cohomology, equipped with the Gauss–Manin connection and Frobenius maps…

Let $f:X\to S$ be a smooth, proper morphism of smooth $k$-varieties, where $k$ is a perfect field of characteristic $p>0$, let $s\in S$ be a closed point, and let…

Hodge-Witt symmetry conjecture. The variety $X$ satisfies Hodge-Witt symmetry. In particular, Ekedahl's conditions hold:

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…