Appendix to: The level 1 weight 2 case of Serre's conjecture - a strategy for a proof

Abstract

In this appendix, we observe that our March preprint on Serre's conjecture was indeed correct: the only "missing argument" follows automatically from a result of Bockle and Ramakrishna. Thus, we get a proof of the level 1 weight 2 case of Serre's conjecture (a result that has been also proved independently by Khare and Wintenberger).

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arXiv:math/0412178v1 [math.NT] 8 Dec 2004
Appendix to: The level 1 weight 2 case of
Serre’s conjecture - a strategy for a proof
Luis V. Dieulefait
Universitat de Barcelona∗
February 1, 2008
1A 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
ˆ ρ|GFthere is at least one that can be extended to GQ.
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 ˆ ρ
and those minimally ramified Barsotti-Tate deformations of ˆ ρ|GFthat can be
extended to the full GQ. (CC)
Let us call R the universal deformation ring of minimally ramified (Barsotti-
Tate at p) deformations of ˆ ρ, and let R′denote a similar minimal universal
deformation ring, but of ˆ ρ|GF.
We know from the results of Taylor that R′is a complete intersection ring.
∗e-mail: ldieulefait@ub.edu
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We will also need the following fact (proved by Bockle and Ramakrishna):
FACT (FF): Let W be the Witt ring of Fq, then R is an W-algebra of
the type:
W[[X1,....Xr]]/(f1,....,fs)
with r ≥ 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
GQ?
Well, if none of the minimal deformations of ˆ ρ|GFdescends to GQ, 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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