4 problems

Let $F$ be a global field of characteristic different from $p$. Let $G$ be a connected reductive group, and let $G'$ be the split group over $F$ whose root datum is dual to that of…

Fontaine–Mazur–Langlands conjecture. There exists an algebraic essentially conjugate self-dual cuspidal automorphic representation $\pi$ of $\operatorname{GL}_m/K$ such that…

Let $Q$ be an auxiliary Taylor–Wiles set, let $\overline{\rho}_{\mathfrak m}$ be the residual representation, let $\mu$ be the specified character, and let $r_{{\mathfrak m},Q}$ be…

Let $F$ be a number field, let $\ell$ be a prime, and let $\rho:\operatorname{Gal}(\overline{F}/F)\to\operatorname{GL}_n(\overline{\mathbb{Q}}_\ell)$ be a Galois representation tha…