Congruences between Hilbert modular forms: constructing ordinary lifts

TL;DR

This paper proves the existence of ordinary modular lifts for certain Galois representations over totally real fields, using a novel approach involving rank 4 unitary groups, with implications for modularity lifting theorems.

Contribution

It introduces a new lifting technique via rank 4 unitary groups to establish the existence of ordinary lifts under mild hypotheses.

Findings

Existence of ordinary lifts for irreducible modular Galois representations.

Improved results on modularity lifting theorems for potentially Barsotti-Tate representations.

Advances on the Buzzard-Diamond-Jarvis conjecture.

Abstract

Under mild hypotheses, we prove that if F is a totally real field, k is the algebraic closure of the finite field with l elements and r : G_F --> GL_2(k) is irreducible and modular, then there is a finite solvable totally real extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each place dividing l. We deduce a similar result for r itself, under the assumption that at places v|l the representation r|_{G_F_v} is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti-Tate representations and the Buzzard-Diamond-Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank 4 unitary groups.