Trianguline lifts of global mod $p$ Galois representations

TL;DR

This paper proves that under certain conditions, irreducible mod p Galois representations of number fields can be lifted to characteristic zero representations that are unramified outside finitely many primes and trianguline at primes dividing p, with extensions to reductive groups.

Contribution

It introduces new conditions under which mod p Galois representations admit characteristic zero lifts that are trianguline at p-adic places and unramified elsewhere, extending to reductive groups.

Findings

Existence of characteristic zero lifts under oddness conditions

Lifts are unramified outside a finite set of primes

Lifts are trianguline at all primes dividing p

Abstract

We show that under a suitable oddness condition, irreducible mod $p$ representations of the absolute Galois group of an arbitrary number field have characteristic zero lifts which are unramified outside a finite set of primes and trianguline at all primes of $F$ dividing $p$. We also prove variants of this result for representations valued in connected reductive groups.