31 problems

In a perfect field $k$, let $X$ be a smooth proper $k$-scheme. Write $\pi_X^i$ for the $i$th Künneth projector of the crystalline cohomology $H^*_{\operatorname{crys}}(X)$, let…

Let $A$ be a discrete valuation ring with finite residue field $k$, let $X$ be a regular scheme projective and flat over $A$ with strict semistable reduction, and put…

Crystalline comparison conjecture. These two complexes are quasi-isomorphic for all $n,m\geq 1$ and all primes $p$; if $A$ is also compact, the homology of either complex is flat o…

Let $k$ be a perfect field, let $X$ be an algebraic variety over $k$, and let $Y$ be a closed subvariety of $X$. Write $W_*$ for a weight filtration on crystalline cohomology, and…

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 $K$ be a $p$-adic field with ring of integers $\mathcal{O}_K$ and residue field $k$, and let $Y$ be a proper and smooth $p$-adic formal scheme over $\mathcal{O}_K$, with generi…

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 $X$ be a variety over a field of characteristic $p>0$, let $W$ denote the ring of Witt vectors, and write $H^2_{\mathrm{crys}}(X/W)_{\mathrm{tor}}$ for the torsion part of its…

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…

Let $X$ be a smooth connected $p$-adic formal scheme over $\mathcal{O}_K$, let ${\mathrm{Spa}}(L)$ be the Shilov point of $X_\eta$, and let $T$ be a $\mathbb{Z}_p$-local system ove…

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 $X$ be a smooth projective variety over a field $k$ of characteristic $p>0$. Its crystalline cohomology $\mathrm{H}^i_{\text{cris}}(X)$ carries Frobenius and decomposes into pa…

Let $X$ be the special fiber of a semistable scheme ${\cal X}$ over the base considered above, let ${\cal W}(s_{\kappa})$ denote the relevant logarithmic crystalline base, and let…

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 projective variety, let $n\ge 0$, and let $\alpha\in H_{dR}^{2n}(X)(n)$. An Ogus cycle is a class satisfying $F_{p,X}(\alpha)=\alpha$ for all sufficiently large…

Sequential period conjecture. The map $\operatorname{per}_{\mathcal{R}}$ is injective. This is the weak analogue of the Grothendieck period conjecture for sequential periods; it pr…

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…

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