13 problems

Let $k$ be a number field and $G$ a transitive permutation group of degree $n$. Write $\mathcal{F}_{\disc,k}(G;X)$ for the set of $G$-extensions $L/k$ with $|\disc(L/k)|\leq X$, wh…

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-$p$…

Let $G$ be a finite group, let $K$ be a number field, and let $\operatorname{inv}_1,\ldots,\operatorname{inv}_n$ denote the multi-invariants used to order $G$-extensions $M/K$, wit…

Localized function-field Boston–Markin conjectures. There exists a positive real number $\delta_q^{G,C}$ such that, as $n_1,\dots,n_{d_\lhd(G)}\to\infty$,

Möbius cancellation conjecture. As $X\to\infty$,

Function-field Boston–Markin conjecture. There exists a regular $G$-extension $K/\mathbb{F}_q(T)$, split completely at infinity, such that the number of ramified primes of…

Boston–Markin conjecture. There exists a totally real tamely ramified $G$-extension $K/\mathbb{Q}$ such that the number of finite primes of $\mathbb{Q}$ ramified in $K$ is…

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-exten…

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-exten…

Let ${\rm K}$ be a number field, and let $\Gamma$ be a uniform pro-$p$ group of dimension $d>2$, so that $\Gamma$ is infinite. A Galois extension ${\rm L}/{\rm K}$ is finitely and…

Let $K$ be a number field, let $G$ be a finite group, and let $V$ be a finite-dimensional faithful representation of $G$ over $K$. Let $N_{G,V,K}(B)$ count isomorphism classes of…

Let $F$ be a global function field of characteristic $p$, and let $G$ be a finite noncyclic abelian $p$-group. Write $Z(F,G;X)$ for the number of Artin-Schreier-Witt extensions…

Revised Malle conjecture. Fixing $N$ and $k$, one has