On the level p weight 2 case of Serre's conjecture

Abstract

This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p representations of G_Q unramified outside p", for the case of trivial character. More precisely, the modularity lifting result needed at a crucial step is the one for the case of a deformation corresponding to a $p$-adic semistable (in the sense of Fontaine) Galois representation attached to a semistable abelian variety, but in Khare's preprint it is applied a lemma only valid for potentially crystalline representations. The completion is easy: both the modularity lifting result applied in Khare's preprint (the one for the case of an abelian variety with potentially good reduction) and the one we need (the one for an abelian variety with bad semistable reduction) can be found in my paper "Modularity of abelian surfaces with Quaternionic Multiplication" (2002): they follow easily from a combination of modularity liftings results a la Wiles and other arguments.