The level 1 weight 2 case of Serre ’ s conjecture - a strategy for a proof

This is a copy of our March 2004 preprint where we attempted to: “prove Serre’s conjecture for the case of Galois representations with 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 and the non-existence of certain p-adic conductor 1 Galois representations”. Since a mistake was discovered during May 2004 in the Galois descent argument, the proof can be considered as incomplete (at some step, we need to extend a Galois representation to the full Galois group of Q, and the reader will see that the existence of an extension is not correctly proved). However, we present this preprint since it describes a strategy for the proof of the small level and weight cases of Serre’s conjecture, which is only incomplete because it depends on the “existence of minimal deformations for residual representations (*)”, which we attempted to deduce from the potential modularity results of Taylor, but we haven’t succeed.