Fontaine–Mazur modularity at the prime 2

Abstract — v1

We prove that every continuous, irreducible, odd two-dimensional $2$-adic representation of $G_{\mathbb Q}$, unramified outside finitely many primes and de Rham at $2$ with distinct Hodge–Tate weights, is modular up to Tate twist. This resolves the odd, regular two-dimensional Fontaine–Mazur conjecture over $\mathbb Q$ at $2$, including all residual representations.