13 problems

Fix a prime number $p$ and let $G\subset W_{2d}$ be a transitive subgroup containing the complex conjugation element. The weighted permutation representation associated to an abeli…

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…

Let $k$ be a Hilbertian field. A finite split embedding problem over $k$ is an epimorphism … where $G$ is finite, $L/k$ is Galois, and there is an embedding…

Folklore conjecture. For any finite group $G$, there exist infinitely many quadratic number fields $K$ such that $K$ possesses a Galois extension with Galois group $G$ unramified a…

Let $G$ be a finite group with a regular realization over $\mathbb Q$, let $P_\ell$ be an $\ell$-Sylow subgroup of $G$, let $m=[G:P_\ell]$, and let $r$ be the branch-point number o…

Let $K\subset\overline{\mathbb Q}$ and suppose that the absolute Galois group $G_K$ is a projective profinite group. The group $G_K$ is profree if it is a free profinite group. Hil…

Let $\mathbb H$ be a modular tower, with levels parametrized by $k$, and let $\mathbb Q$ denote the rational field. A variety is of general type when some multiple of its canonical…

Let $\ell$ be a prime, $k\geq 0$, and $d$ a fixed dimension. A cyclotomic point means a rational point defined over a cyclotomic extension, and $D_{\ell^{k+1}}$ denotes the dihedra…

Strong-torsion bound conjecture. Some expression in $r$ and $m$, independent of the prime of good reduction $v$, bounds $p^u$.

High-level modular-tower conjecture. At sufficiently high levels, there are no $K$-points on a modular tower. Moreover, sufficiently high levels are algebraic varieties of general…

Let $p>2$ be a prime, let $G=\operatorname{GL}_2(\mathbb{F}_p)$, and let $(G,I,p)$ denote the realizability problem with inertia subgroup $I$. An inertia candidate is a subgroup…

Let $p$ be a prime and let $G$ be a nontrivial finite group. Write $G^{ab}$ for the abelianization of $G$, let $p(G)$ be the normal subgroup generated by the elements of $p$-power…

Rigid-triple conjecture. The triple $(C_0,C_1,C_{\infty})$ is strictly rigid in $\widehat{G}(\mathbb{F}_{\ell})$: the equation