The level $1$ weight $2$ case of Serre's conjecture

Abstract

We prove Serre's conjecture for the case of Galois representations of Serre's weight $2$ and level $1$. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of $p$-adic Barsotti-Tate conductor $1$ Galois representations proved in [Dieulefait, L.: Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture. J. Reine Angew. Math. 577 (2004), 147-151].