Remark: Family 010. Fontaine–Mazur modularity at the prime 2 and 2 -adic pro-modularity

Every continuous odd absolutely irreducible two-dimensional $2$-adic representation of $G_{\mathbb{Q}}$ unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at $2$ with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.