21 problems

Let $p$ be an odd prime and let $t$ be a positive integer. A number field $K$ is $p$-rational if, for the set $S$ of primes of $K$ lying above $p$, the Galois group of the maximal…

Gras's conjecture. For $p\gg 0$, one has

Let $p$ be a prime, let $G$ be a finite group, let $P\in\operatorname{Syl}_p(G)$, and let $\alpha\in\mathbb{Z}^{+}$. The $p$-rationality level of an irreducible character is determ…

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Define … and let $Q\in\operatorname{Syl}_p(K)$. Call a character almost $p$-rational when it h…

Hajir–Maire's conjecture. There exist a totally imaginary number field $K_0$ and a cyclic extension $K/K_0$ of degree $m$ such that $K$ is $p$-rational; moreover, the statement is…

Let $G$ be a finite group, let $p$ be a prime, and let $P\in\operatorname{Syl}_p(G)$. Set … Call a character almost $p$-rational when it has degree coprime to $p$ and $p$-rationali…

Let $p$ be a prime and let $S$ be a finite non-abelian simple group whose order is divisible by $p$. Simple-group reduction conjecture. The following assertions hold. (a) Let $X$ b…

Let $G$ be a finite group and $p$ a prime. Let $B_0(G)$ denote the principal $p$-block of $G$, and write $c(\theta)_p=p^r$ for the $p$-rationality level of a character $\theta$. Hu…

Let $G$ be a finite group and $p$ a prime. For $\theta\trust in\operatorname{Irr}(G)$, call $\theta$ of degree coprime to $p$ when $p\nmid\theta(1)$, and write $c(\theta)_p=p^r$ fo…

Let $p$ be a prime and $G$ a finite group. Let $B$ be a $p$-block of $G$ with defect group $D$, let $b$ be its Brauer correspondent in $\mathbf{N}_G(D)$, and let $\mathcal{H}_B$ be…

Let $p$ be a prime, $G$ a finite group, and let $B$ be a $p$-block of $G$ with defect group $D$. Let $u$ be the $p$-adic valuation, let $c(rho)$ denote the conductor of the values…

Let $K=\mathbb{Q}(\sqrt{D})$ be a fixed real quadratic field, let $u_K$ be its fundamental unit, and let $p$ be a finite prime. Condition $(\mathbf{W})$ is … More generally, for so…

Let $p$ be a prime, let $S$ be a finite nonabelian simple group, and let $S\trianglelefteq X\leq\operatorname{Aut}(S)$, where $X$ is almost simple and $X/S$ is a $p$-group. Let…

Let $p$ be a prime, let $G$ be a finite group, and let $\chi$ be an irreducible $p'$-degree character of $G$. The $p$-rationality level of a character value is defined using the…

Let $p$ be a prime, let $G$ be a finite group, let $\alpha\in\mathbb{Z}^{\geq 2}$, and let $M$ be a subgroup of $G$ of $p'$-index. Continuity under p'-index subgroups. The group…

Let $p$ be a prime, $G$ a finite group, and let $\alpha\in\mathbb{Z}^{\geq 2}$. The $p$-rationality level of a character is the integer determined by the $p$-part of its conductor.…

Let ${\mathcal K}$ be the set of totally real number fields. For $K\in{\mathcal K}$, let $D_K$ be its discriminant, let ${\mathcal T}_K$ be the relevant $p$-torsion group, and let…

Let $N\geq 2$ and let $p$ be a prime with $p\nmid N$. Let $\psi_N$ be a character of $\mathbb{Q}(\mu_N)$ of order $N$, and let ${\mathfrak p}_N$ be a prime above $p$ in…

Let $K/\mathbb{Q}$ be a real Galois extension with Galois group $G$, and let $\eta\in E_K$ be a Minkowski unit, meaning a unit generating a sub-$G$-module of finite index in $E_K$.…

Fix a prime $p\geq 5$ and an integer $m>1$ not divisible by $p$. Let $K_0$ be a $p$-rational quadratic imaginary field, and let $\mathcal F_m(K_0)$ be the family of cyclic extensio…

Let $p$ be a prime and let $m\geq 1$ be an integer coprime to $p$. A number field is $p$-rational when it has the standard $p$-rationality property used in the paper. Cyclic p-rati…