11 problems

Let $S$ be a finite set of places, and let $\rho$ be an irreducible $\ell$-adic Galois representation of $\mathbb Q$ that is unramified outside $S$ and de Rham locally at $\ell$. F…

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

Let $F$) be a number field and let $\overline{\rho}\colon G_F\to\operatorname{GL}_2(\mathbb{F}_p)$ be a continuous absolutely irreducible residual Galois representation whose image…

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 $p$ be a prime, and let $G_F$ be its absolute Galois group. For a tangential base point $0_v$ supported at $0$, write … for the subspace of regular f…

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 $p$ 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…

Boston's strengthening of the unramified Fontaine–Mazur conjecture. The universal unramified deformation $\rho^{\mathrm{unr}}$ has finite image.

Let $F$ be a number field and let $\rho:\operatorname{Gal}(\overline{F}/F)\rightarrow GL_n(\overline{\mathbb{Q}}_l)$ be a continuous irreducible representation unramified outside f…