9 problems

Reduced Boston conjecture. 1. If $G$ is a uniform $p$-adic analytic pro-$p$ group of simple type in characteristic zero and is hereditarily just-infinite, then there is no pro-$p$…

Unramified Fontaine–Mazur conjecture. Every continuous homomorphism

Geometric-to-strict compatibility conjecture. Any semisimple geometric representation

Universality conjecture. Any irreducible $\overline{\mathbb{Q}}_p$-representation of $G_F$ that is almost everywhere unramified and de Rham at places above $p$ can be established a…

Let $F$ be a number field, let $X$ be a smooth variety over $F$, and let ${\mathbb L}$ be an irreducible $\overline{\mathbb{Q}}_p$-local system on $X$. Assume that there is a finit…

Let $K$ be a number field, and let … denote the Galois group of the maximal unramified extension of $K$. Fontaine–Mazur's unramified conjecture. Every continuous Galois representat…

Let $K$ be a number field and $p$ a prime. Let $K^{ur}(p)$ be the maximal unramified pro-$p$ extension of $K$, and set … A quotient is uniform if it is a uniform pro-$p$ group. Unr…

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

Let $F$ be a number field and let … be an irreducible geometric Galois representation, meaning that it is almost everywhere unramified and de Rham at every place above $\ell$. Font…