Serre's modularity conjecture: The level one case

Abstract

We prove the level one case of Serre's conjecture. Namely, we prove that any continuous, odd, irreducible representation $\stackrel{̲}{\rho }:G_{\mathbb{Q}}\to {\mathrm{GL}}_2(\stackrel{̲}{{\mathbb{F}}_p})$ which is unramified outside $p$ arises from a cuspidal eigenform in $S_k({\mathrm{SL}}_2(\mathbb{Z}))$ for some integer $k\ge 2$. 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