32 problems

A field $K$ is Hilbertian if it satisfies Hilbert's irreducibility property. The inverse Galois conjecture for Hilbertian fields. Over every Hilbertian field $K$, the inverse Galoi…

A finite group $G$ is admissible over a field $F$ if there exists a central division algebra over $F$ containing a $G$-Galois field extension as a maximal subfield. Equivalently, t…

Let $k$ be a field, let $x_1,\dots,x_k$ be variables, and let $G$ be a finite group acting by permutation on these variables. Noether's rationality conjecture. The invariant field…

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…

Weak Main Conjecture. There cannot be $K$ points at every level of a modular tower. Consequently, regular realizations of all the $G_k$ over $K$ require an unbounded number of bran…

The higher-dimensional inverse Galois conjecture. The elements $t_{d+1}(X/Y)$, as $X\to Y$ ranges over all such torsors, generate

Let ${\bf F}_q$ be a fixed finite field, and let $G$ be a simple algebraic group of adjoint type defined over ${\bf F}_q$. Thompson's conjecture. There exist regular Galois realiza…

Let $G$ be a finite group, let $p$ be a prime divisor of $|G|$, let ${\underline C}_1$ be an unordered $r$-tuple of conjugacy classes of $G$ of order prime to $p$, and let $G_n$ an…

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 $G$ be a finite group, and let $K$ range over quadratic number fields. An extension of $K$ is unramified at all primes when it is unramified at every finite and infinite prime…

Let $G$ be a non-trivial finite étale tame $F$-group scheme, and let $H:BG(F)\to\mathbb{R}_{>0}$ be a height. Define the invariants $a(H)$ and $b(H)$ as in Definition 4.1. Quantita…

Let $G$ be a finite group, and write $\operatorname{pd}_{\mathbb{Q}}(G)$ for its parametric dimension over $\mathbb{Q}$. Parametric-dimension conjecture. The only groups $G$ with ……

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…

Essential dimension conjecture. When ${\rm{gd}}_k G$ is finite, one should have

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…

Beckmann–Black lifting conjecture. Every $H$-extension of a number field $K$ is a specialization of a $K$-regular $H$-extension of $K(t)$.

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…

Torsion conjecture. The conclusion of the theorem asserting the existence, for every $k\geq 0$, of such a $\mathbb Q$-cyclotomic point on a hyperelliptic Jacobian of one fixed dime…

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…

Fried's conjecture. Every set $\{C_1,\dots,C_r\}$ of non-trivial conjugacy classes of $G$ that is rational and whose elements generate $G$ occurs as the inertia canonical conjugacy…