16 problems

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category with $p$-orbit category…

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, let $G$ be a finite group, and let $\mathcal{O}_G$ be its $p$-orbit c…

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, set $\Delta=\Delta_d=(d+1)(d-3)$ and $K=\Q(\sqrt{\Delta})$. Let $\mathcal O_\Delta$ and $\mathcal O_K$ denote the quadratic order of discriminan…

Fix a positive integer $d\neq 3$, let $\Delta_d=(d+1)(d-3)$, and write $\Delta_d=f^2\Delta_0$, where $\Delta_0=\operatorname{disc}(\mathcal O_{K_d})$ and $K_d=\Q(\sqrt{\Delta_d})$.…

Suppose the semisimple local Langlands correspondence $bbLL_{bbG}$ exists. Let $bbsigma$ be an automorphism of $bbC$ fixing $bqsqrt q$, and let $(bbM,bbrho)$ be a supercuspidal-sup…

Galois-invariant normal Sylow subgroup conjecture. Let $G$ be a finite group and let $q$ be a prime. Then $G$ has a normal Sylow $q$-subgroup if and only if, for any $p\neq q$ that…

Let $G$ be a finite group, let $p$ be a prime, and let $B$ be a $p$-block of $G$ with defect group $D$. Let $b$ be the Brauer correspondent of $B$ in $N_G(D)$. Let $\mathcal H$ be…

Fix a prime $p$ and a finite group $G$. Let $\mathcal{H}$ be the Galois automorphism group associated with the $p$-modular system in the source. Let $\mathcal{W}(G)$ be the set of…

Fix a prime $p$ and a finite group $G$. Let $\mathcal{H}=\operatorname{Gal}(\mathcal{K}/\mathbb{Q}_p)$ be the finite abelian group of Galois automorphisms associated with the $p$-m…

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Let $B_0$ be the principal $p$-block of $G$ and $b_0$ the principal $p$-block of $N_G(P)$…

Let $p\in\{2,3\}$. Let $G$ be a finite group and let $B$ be a $p$-block of $G$ with nontrivial defect group $D$. Let $\sigma_1$ be the Galois automorphism introduced in the paper,…

Let $G$ be the reductive group in the source, let $K$ be the chosen model of a maximal compact subgroup, and let $\Pi$ be an irreducible cohomological cuspidal automorphic represen…

Let $G$ be the reductive group in the source, let $K$ be the chosen model of a maximal compact subgroup, and let $M$ be a rational $G$-representation. For a factorizable automorphi…

Let $G$ be a finite group, let $p$ be a prime dividing $|G|$, let $D$ be a Sylow $p$-subgroup of $G$, and let $N_G(D)$ be the normalizer of $D$ in $G$. Let $\operatorname{Irr}(G)$…

Triangle-curve faithfulness conjecture. The absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the set of isomorphism classes of unmark…

Faithful-action conjecture. There is a minimal surface of general type $S$ such that $S$ and its conjugate $S^{\phi}$ have non-isomorphic fundamental groups. In particular, $S$ and…