The odd regular two-dimensional Fontaine-Mazur conjecture over Q at p = 2

Theorem 1.1: every continuous, irreducible, odd $r:G_{\mathbb Q}\to\mathrm{GL}_2(\overline{\mathbb Q}_2)$ unramified outside finitely many primes and de Rham at $2$ with distinct Hodge-Tate weights is a Tate twist of the representation of a classical cuspidal eigenform; no residual hypothesis. With the odd-prime results this gives the regular odd two-dimensional case over $\mathbb Q$ at every prime. It does not address equal Hodge-Tate weights, even representations, higher dimension or other number fields, so the general Fontaine-Mazur conjecture stays open. Companions prove 2-adic pro-modularity without a de Rham hypothesis (October 4) and Emerton's dimension conjecture at $p=2$ (October 5), entered separately.

Theorem 1.1: every continuous, irreducible, odd $r:G_{\mathbb Q}\to\mathrm{GL}_2(\overline{\mathbb Q}_2)$ unramified outside finitely many primes and de Rham at $2$ with distinct Hodge-Tate weights is a Tate twist of the representation of a classical cuspidal eigenform; no residual hypothesis. With the odd-prime results this gives the regular odd two-dimensional case over $\mathbb Q$ at every prime. It does not address equal Hodge-Tate weights, even representations, higher dimension or other number fields, so the general Fontaine-Mazur conjecture stays open. Companions prove 2-adic pro-modularity without a de Rham hypothesis (October 4) and Emerton's dimension conjecture at $p=2$ (October 5), entered separately.