Appendix to: The level 1 weight 2 case of Serre's conjecture - a strategy for a proof
Luis V. Dieulefait
% In this appendix, we complete the proof of the level 1 weight 2 case of Serre's conjecture given in our previous preprint, % just answering the sphinx question. %Mathematics Subject Classification: 11F80, 11F46 %
math0412178.tex
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\title{Appendix to: The level 1 weight 2 case of Serre's conjecture - a strategy for a proof}
\author{Luis V. Dieulefait\\ Universitat de Barcelona
  \thanks{
e-mail: ldieulefait@ub.edu
}} 


\begin{document}

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%\noindent Running head:
%\begin{abstract}
% In this appendix, we complete the proof of the level 1 weight 2 case of Serre's conjecture given in our previous preprint,
% just answering the sphinx question.
%Mathematics Subject Classification: 11F80, 11F46
%\end{abstract}
%\newpage

\section{A fact, a question, and its answer}
(we follow the notation of the preprint)

 As explained in the preprint, to complete the proof our task is to show that, among the minimal (modular) Barsotti-Tate p-adic deformations of 
 $\hat{\rho}|_{G_F}$ there is at least one that can be extended to $G_\Q$.\\

%(We follow the notation of the preprint). Recall that the only missing point was to show that there 
%exists a Barsotti-Tate minimal p-adic deformation of $\hat{\rho}$, and we use the fact that Barsotti-Tate  minimal modular %deformations exist
% for the restriction $\hat{\rho}|_{G_F}$. 
The field $F$ is known to be unramified at $p$ and we can choose it (as explained in the preprint, using solvable base-change and Sylow theorems) so that its degree over $\Q$ is prime to $p$.\\

 With these conditions, as we already explained, there is a one-to-one correspondence between minimally ramified Barsotti-Tate deformations of $\hat{\rho}$ and  those minimally ramified Barsotti-Tate deformations of $\hat{\rho}|_{G_F}$ that can be extended to the full $G_\Q$. (CC)\\
 
 Let us call $R$ the universal deformation ring of minimally ramified (Barsotti-Tate at $p$) deformations of $\hat{\rho}$, and let $R'$ denote a similar minimal universal deformation ring, but of $\hat{\rho}|_{G_F}$.\\
 We know from the results of Taylor that $R'$ is a complete intersection ring.\\
  We will also need the following fact (proved by Bockle and Ramakrishna):\\
 
 FACT  (FF): Let $W$ be the Witt ring of $\F_q$, then  $R$ is an $W$-algebra of the type:
 
 $$  W [[ X_1, .... X_r]] / (f_1, ...., f_s) $$
 
 with $r \geq s$.\\
 
 Now, let us answer the question: Why in the complete intersection ring $R'$ there is at least one p-adic deformation that survives when we descend to $G_\Q$?\\
 Well, if none of the minimal deformations of $\hat{\rho}|_{G_F}$  descends to $G_\Q$, then using the correspondence (CC) we see that $R$ is too small to match with (FF).\\
 This concludes the proof, so existence of minimal deformations, and Serre's conjecture in the level 1 weight 2 case follow.\\
  
 
 
 
 
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