All Questions

Let $G$ be an arbitrary open group of $SL_2(\mathbb{Z}_p)$ and $K$ be a finite extension of $\mathbb{Q}_p$. Can we construct a Galois extension field $E$ of $K$ such that $\text{Gal}(E/K)\cong G$? ...

What's the current state of the Breuil-Mezard conjecture? Has the original version (from the 2002 paper) been solved in its entirety? What are some of the new directions being explored?