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

Abstract

We study irreducible odd mod p Galois representations $\overline{\mathit{\rho }}:\mathrm{Gal}(\bar{F}∕F)\to G({\bar{\mathbb{F}}}_p)$, for F a totally real number field and G a general reductive group. For $p{\gg }_{G,F}0$, we show that any $\overline{\mathit{\rho }}$ 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.