Number theory · Oct 6, 2026
Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of 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.
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