55 problems

Let $F$ be a field containing a root of unity of order $p$, and let $G_F(p)$ be its maximal pro-$p$ Galois group. Let $\mathfrak{B}$ be the class of standard building blocks and le…

Let $p$) be a prime, let $\bar G$ be a pro-$p$ group, let $n$ be a positive integer, and let $\bar\omega\in H^n(\bar G,\mathbb{F}_p)$. For a field $F$ containing a root of unity of…

Let $P$ be a pro-$p$ group, and write … Let $\varphi^2:H_{\operatorname{cont}}^{2}(P)\to H_{\operatorname{disc}}^{2}(P)$ be the comparison map. Prasad's conjecture. For every $p$-a…

Let $K=\mathbb{Q}_q$ be unramified over $\mathbb{Q}_p$, let $\overline{k}$ be an algebraic closure of its residue field, and let $P=\operatorname{Gal}(K^s/K^\mathrm{tr})$ be the wi…

Let $p>2$ be a prime and let $F$ be a field containing a primitive $p$-th root of unity. Write $F(p)$ for the maximal $p$-extension of $F$, let $G=\operatorname{Gal}(F(p)/F)$, and…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let positive-dimensional mean having positive Hausdorff dimension as a closed subgroup of…

Let $G$ be a just-infinite pro-$p$-group of finite width, meaning a just-infinite pro-$p$-group whose finite-index subgroup growth layers have uniformly bounded width. Zel'manov's…

Ozaki's non-freeness conjecture. The group $\mathcal{G}(k^\mathrm{c}_\infty)$ is not a non-abelian free pro-$p$-group.

Open Golod–Shafarevich subgroup conjecture. The group $\operatorname{Gal}(L/K)$ contains an open Golod–Shafarevich pro-$p$ subgroup.

Boston's virtually Golod–Shafarevich conjecture. The group $\operatorname{Gal}(L/K)$ contains an open pro-$p$ subgroup $G$ such that $d(G)$ and $r(G)$ are finite and

Reduced Boston conjecture. 1. If $G$ is a uniform $p$-adic analytic pro-$p$ group of simple type in characteristic zero and is hereditarily just-infinite, then there is no pro-$p$…

Demuškin maximal pro-$p$ Galois group conjecture. If $G_F(p)$ is a pro-$p$ Demuškin group of rank at least $3$, then $F$ is arithmetically Demuškin.

Demuškin-to-$p$-adic type conjecture. If $\mathcal{G}_F(p)$ is a Demuškin cyclotomic pro-$p$ pair of rank at least $3$, then $\mathcal{G}_F(p)$ has $p$-adic type.

Elementary Type Conjecture in Lego form. Every cyclotomic pro-$p$ pair of Galois type and of finite rank is in $\operatorname{ET}_p$.

Efrat's arithmetical Elementary Type Conjecture. There is such a decomposition where, for each $1\leq i\leq n$, one of the following holds: $G_i\cong\mathbb{Z}_p$; $G_i$ is the dec…

Massey vanishing conjecture. $G_K$ satisfies the Massey vanishing property.

Let $p$ be a prime, let $T_p$ be the rooted $p$-ary tree, and let $W_p$ be its full automorphism group. A pro-$p$ group is just-infinite if all its non-trivial closed normal subgro…

Shalev's probabilistic identity conjecture. If $G$ satisfies some probabilistic identity $w$, then $G$ satisfies some identity.

Let $G$ be a saturable pro-$p$ group with associated $\mathbb{Z}_p$-Lie algebra $\mathfrak{g}$. Here $\operatorname{H}^{*}(G;\mathbb{F}_p)$ and…

Let $K$ be a number field, let $S_p$ be the set of $p$-adic primes of $K$, and let $\Gamma_{S_p}=\operatorname{Gal}(K_{S_p}/K)$, where $K_{S_p}$ is the maximal pro-$p$ extension of…

Weak unramified Fontaine–Mazur conjecture. Every continuous homomorphism

Let $\Gamma=(\mathcal{V},\mathcal{E})$ be a special digraph, let $G$ be the oriented pro-$p$ right-angled Artin group associated to $\Gamma$, and let $q$ be a $p$-power with…

Finite-codimension conjecture. If $G$ is an almost simple uniform pro-$p$ group, then every non-zero prime ideal of $KG$ has finite codimension.

Let $p>3$ be a prime, and let $K$ be an imaginary quadratic field whose class group has $p$-rank two. The Zassenhaus type is the pair $(i,j)$ occurring in the Zassenhaus polynomial…

Let $p$ be a prime, let $K$ be a number field, and let $S$ be a finite set of places of $K$ not dividing $p$. Let $K_S$ be the maximal pro-$p$ extension of $K$ unramified outside…