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

Abstract

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

Minor clarifications to previous version. Final version, to appear in DMJ. arXiv admin note: text overlap with arXiv:1810.05803