Trianguline lifts of global mod $p$ Galois representations

Abstract

We show that under a suitable oddness condition, irreducible mod p representations of the absolute Galois group of a totally real 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 and also for representations of the absolute Galois group of a CM field.