Serre's modularity conjecture: The level one case

We prove the level one case of Serre's conjecture. Namely, we prove that any continuous, odd, irreducible representation ρ̲:GQ→GL2(Fp̲) which is unramified outside p arises from a cuspidal eigenform in Sk(SL2(Z)) for some integer k≥2. The proof relies on the methods introduced in an earlier joint work with J.-P. Wintenberger [31] together with a new method of weight reduction