On 2-dimensional 2-adic Galois representations of local and global fields

Abstract

We describe the generic blocks in the category of smooth locally admissible mod-$2$ representations of ${GL}_2\left({\mathbb{Q}}_2\right)$. As an application we obtain new cases of the Breuil–Mézard and Fontaine–Mazur conjectures for $2$-dimensional $2$-adic Galois representations.