From potential modularity to the level 1 weight 2 case of the conjectures of 
Fontaine-Mazur and Serre 


Abstract: I will explain how from the results of Taylor of potential 
modularity of Galois representations one can deduce a result of "existence of 
a family" containing certain p-adic Galois representations, and deduce from 
this the level 1 weight 2 case (and other cases of small ramification) of the 
conjecture of Fontaine-Mazur-Langlands (FML), and also the FML conjecture 
for "elliptic curves". Finally, using again potential modularity and lowering 
the level for Hilbert modular forms I will explain how a result of "existence 
of minimal deformations" can be obtained, at least in the semistable case, 
thus reducing the proof of some semistable cases of Serre's conjecture  to the 
proof of the FML conjecture in the same case, therefore in particular proving 
Serre's conjecture in the level 1 weight 2 case and other cases of small 
ramification (remark: a similar result has been proved independently by Khare 
and Wintenberger).