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

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