Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

Abstract

We study irreducible odd mod p Galois representations ρ¯:Gal(F‾∕F)→G(F‾p), for F a totally real number field and G a general reductive group. For p≫G,F0, we show that any ρ¯ that lifts locally, and at places above p to de Rham and Hodge–Tate regular representations, has a geometric p-adic lift. We also prove non-geometric lifting results without any oddness assumption.