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

TL;DR

This paper characterizes the structure of certain mod 2 representations of GL_2(Q_2) and applies these results to verify new instances of conjectures relating Galois representations and automorphic forms.

Contribution

It describes the generic blocks in the category of smooth locally admissible mod 2 representations of GL_2(Q_2) and uses this to prove new cases of Breuil--Mézard and Fontaine--Mazur conjectures.

Findings

Classification of generic blocks in the representation category

New cases of Breuil--Mézard conjecture verified

New cases of Fontaine--Mazur conjecture established

Abstract

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.