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

## TL;DR

This paper investigates conditions under which irreducible mod p Galois representations over totally real fields can be lifted to geometric p-adic representations, expanding understanding of their deformation theory and lifting properties.

## Contribution

It establishes new criteria for geometric lifts of Galois representations and proves non-geometric lifting results without oddness assumptions.

## Key findings

- Any locally liftable irreducible mod p Galois representation with certain local conditions admits a geometric p-adic lift.
- Proves non-geometric lifting results for Galois representations without the oddness restriction.
- Provides a framework connecting deformation theory, Selmer groups, and lifting of Galois representations.

## Abstract

We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, 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.