On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q)

@article{Khare2009OnSC,
  title={On Serre's conjecture for 2-dimensional mod p representations of Gal( Q=Q)},
  author={Chandrashekhar B. Khare and J.-P. Wintenberger},
  journal={Annals of Mathematics},
  year={2009},
  volume={169},
  pages={229-253}
}
We prove the existence in many cases of minimally ramied p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre’s conjectures in low levels and weights. For instance we prove that there are no irreducible (p;p) type… 
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