We describe the generic blocks in the category of smooth locally admissible mod $2$ representations of $\mathrm{GL}_2(\mathbb{Q}_2)$. As an application we obtain new cases of Breuil--M\'ezard and Fontaine--Mazur conjectures for $2$-dimensional $2$-adic Galois representations.
On a relation of overconvergence and $F$-analyticity on $p$-adic Galois representations of a $p$-adic field $F$
Similarity of quadratic forms over global fields in characteristic 2
On the automorphy of $l$-adic Galois representations with small residual image
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