Lifting modular mod \(l\) representations (Q1331677)

The article extends previous work of the authors and others. The main result is that once a modulo \(l\) (odd prime) Galois representation has a modular lift to characteristic zero, then one can find another such lift with prescribed restriction to the decomposition groups at primes \(p\neq l\). The proof uses a careful analysis of the possible local representations and the Jacquet-Langlands correspondence.