Multiplicity one of regular Serre weights

摘要

We consider a potentially Barsotti-Tate deformation problem of a modular Galois representation. By constructing a Diamond-Taylor-Wiles system, we prove an R = T theorem and a multiplicity one result in characteristic 0. Applying this result, we then prove a multiplicity one result in characteristic p, which provides certain evidence for a conjecture of Breuil.