40 problems

Fontaine's ramification conjecture. The group $G_K^{(\mu)}$ should act trivially on $\mathrm{H}^i_{\text{\'e}t}(X_{\overline{K}}, \mathbb{Z}/p^n\mathbb{Z})$ when

Let $\Lambda[\pi(0)]$-modules be modules associated with the break-zero part of the relevant $p$-adic representation-theoretic filtration. A module is pure of break zero when its S…

Let $E$ be the local field under consideration and let $G_E$ denote its absolute Galois group. Let $\rho$ be a representation of $G_E$ with finite local monodromy. For a positive i…

Let ${\mathscr F}$ be a constructible étale sheaf of flat $\Lambda$-modules on $\eta$, and let $r>1$ be a critical slope of ${\mathscr F}$. Assume that … Thus $r$ is the unique cri…

Let $X$ be a separated scheme of finite type over a perfect field $F$, let $U\subset X$ be dense, connected, smooth, and purely $d$-dimensional, let $\ell$ be invertible in $F$, an…

Swan trace formula conjecture. Then

Let $U$ be a smooth connected scheme of dimension $d$ over a perfect field $F$, and let $\mathcal F$ be a smooth $\overline{\mathbb F}_\ell$-sheaf on $U$. Swan class conjecture. (1…

Serre's conjecture. The function $a_G$ is a character of the group $G$. This is attributed in the source to Serre and concerns the character-theoretic integrality of local fixed-po…

Let $f:V\to U$ be a finite étale morphism of connected separated and smooth schemes of finite type purely of dimension $d$ over a perfect field $F$, and let $\sigma$ be a non-trivi…

Kuhlmann–Rzepka's characterization conjecture. A henselian field is a roughly deeply ramified field if and only if all of its algebraic extensions are independent defect fields.

Let $G$ be a connected reductive group over $F$ that splits over $F^t$, let $\varphi$ be a positive-depth Langlands parameter for $G(F)$, and let $\Pi(\varphi)$ be its associated…

Strict inequality conjecture. Assume $p=2$. For any irreducible orthogonal Galois representation $\rho$ of $G$ which is not tame, one has

Arboreal Sen Conjecture for PCF maps. If $p \mid d$, then the iterated extension

Suppose that $X$ is a surface, and let $D$ and the invariant $s_x$ be as in the preceding notation: for a closed point $x$ of $D$, $s_x$ is the integer defined from the ramificatio…

Suppose that $X$ is purely of dimension $d$ and that $D$ has simple normal crossings. Let $I'subset I$ be a subset containing $I_{\mathrm{T},\mathcal{F}}$, and let … Assume that th…

Let $p$ be a prime, let $G e mathbb{Z}/p^nrtimesmathbb{Z}/m$ be non-abelian with $pnmid m$, and let $k[[z]]/k[[t]]$ be a local $G$-extension whose $mathbb{Z}/p^n$-subextension has…

Let $M$ be an irreducible overconvergent $F$-isocrystal on $\operatorname{Spec}(F)$ with nonnegative slopes. Let $M^{u-r}$ be its unit-root subcrystal, and let $\eta$ be its first…

Let $K'/K$ be a geometric, purely inseparable, untwisted extension of geometric discrete valuation fields with finite exponent. Write $e$ for its ramification index, $f$ for its du…

Esnault–Kerz conjecture. One has

Esnault–Kerz conjecture. These conditions are equivalent to requiring that, for every birational morphism $f:X'\rightarrow X$ such that $X'$ is smooth over…

Oort conjecture revisited. Local cyclic extensions are liftable in towers: given the local $G$-extension $k[[z]]/k[[t]]$ and a characteristic-zero lift $R[[S]]/R[[T]]$ of a subexte…

Dihedral weak local Oort conjecture. The group $D_{2^n}$ is a weak local Oort group for $2$.

Equality conjecture. The two generalized ramification functions satisfy

Polynomial ramification criterion. For every positive integer $b$, there exists a polynomial $P_b\in\mathbb{Z}[x_1,\dots,x_{b+1}]$ such that $f$ is $b$-ramified if and only if…

Ramification conjecture. Under these assumptions, if $G_{M_i,\,\log}^{t+}$ acts trivially on $\mathscr{G}_{\overline{\eta_i}}$ for every $i$, then $G_{K,\log}^{t+}$ acts trivially…