13 problems

Let $\Gamma\cong\mathbb Z/3$ and let a 2-select group mean a group in the class defined by Boston and Bush: a quotient of a suitable 2-special group by the normal $\Gamma$-invarian…

Fix a number field $k$ with no nontrivial everywhere unramified extension, a finite group $\Gamma$, and conjugacy classes $\gamma_v$ at the infinite places. Let $n$ be the number o…

Let $K$ be a cyclic cubic extension of $\mathbb Q$, necessarily split at infinity, and let $K^{\operatorname{un},2}$ be the composition of all finite Galois extensions of $K$ that…

The first tower-length conjecture. If

Let $p>3$ be a prime, and let $K$ be an imaginary quadratic field whose class group has $p$-rank two. Set … Choose a basis $x,y$ of $H^1(G,\mathbb{Z}/p\mathbb{Z})$, and let…

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…

The $A(8)=3/2$ conjecture. It is conjectured that

Tower ground state conjecture. The fields with discriminants $d\in\{-225\,299,-343\,380,-423\,476,-486\,264\}$ of type $\mathrm{F}.7$,…

Extremal root path conjecture. The root path $(\pi^{i-1}(G)\to\pi^i(G))_{1\leq i\leq c-2}$ consists of edges of maximal step size, equal to the nuclear rank: for every…

Let $B$ be a positive bound, let $d$ range over fundamental discriminants with $0<d<B$, and let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic field. Consider the groups…

Let $B$ be a positive bound, let $d$ range over fundamental discriminants with $0<d<B$, and let $K=\mathbb{Q}(\sqrt{d})$ be a real quadratic field. Consider the groups…

Let $p$ be a prime. A finite $p$-group has a minimal number of generators $d(G)$ and a minimal number of relations $r(G)$. Shafarevich's conjecture. There cannot be any sequence of…

Let $k$ be a number field and let $l$ be a prime. Write $\pi_1(\operatorname{Spec}({\cal O}_k))(l)$ for the maximal pro-$l$ quotient of the étale fundamental group of…