add bestiary

Author: kbuzzard

blueprint/src/chapter/ch02reductions.tex

@@ -99,4 +99,4 @@ \section{Reduction to two big theorems.}
 Indeed, if Fermat's Last Theorem is false then there is a Frey package $(a,b,c,p)$ by~\ref{Frey_package_of_FLT_counterex}, contradicting Corollary~\ref{no_Frey_package}.
 \end{proof}
 
-The structure of the rest of this document is as follows. In Chapter 3 we develop some of the basic theory of elliptic curves and the Galois representations attached to their $p$-torsion subgroups. We then apply this theory to the Frey curve, deducing in particular how Mazur's result on torsion subgroups of elliptic curves implies Theorem~\ref{Mazur_on_Frey_curve}, the assertion that $\rho$ cannot be reducible. In Chapter~\ref{ch_overview} we give a high-level overview of our strategy to prove that $\rho$ cannot be irreducible, which diverges from the original approach taken by Wiles; one key difference is that we work with the $p$-torsion directly rather than switching to the 3-torsion. In this chapter we also state several more results from the 1980s and before which are necessary ingredients in our proof. We then begin the journey towards the proof of the modularity lifting theorem which provides the key ingredient in our approach.
+The structure of the rest of this document is as follows. In Chapter 3 we develop some of the basic theory of elliptic curves and the Galois representations attached to their $p$-torsion subgroups. We then apply this theory to the Frey curve, deducing in particular how Mazur's result on torsion subgroups of elliptic curves implies Theorem~\ref{Mazur_on_Frey_curve}, the assertion that $\rho$ cannot be reducible. In Chapter~\ref{ch_overview} we give a high-level overview of our strategy to prove that $\rho$ cannot be irreducible, which diverges from the original approach taken by Wiles; one key difference is that we work with the $p$-torsion directly rather than switching to the 3-torsion. We also give a precise statement of the modularity lifting theorem which we will use. Finally, in Chapter~\ref{ch_bestiary} we give a collection of theorem statements which we shall need in order to push our strategy through. All of these results were known in the 1980s or before. This chapter is incoherent in the sense that it is just a big list of apparently unrelated results. As our exposition of the proof expands, the results of this chapter will slowly move to more appropriate places. The chapter is merely there to give some kind of idea of the magnitude of the project.

blueprint/src/chapter/ch03frey.tex

@@ -412,19 +412,3 @@ \section{Hardly ramified representations}
 \begin{proof} Follows from theorem~\ref{Frey_curve_reducible_structure}, corollary~\ref{Frey_curve_no_trivial_submodule}
   and corollary~\ref{Frey_curve_no_trivial_quotient}.
 \end{proof}
-
-{\bf TODO} there's some commented-out code in the LaTeX file which might be useful here.
-
-%\begin{theorem}\label{i_am_not_going_to_give_any_uses} A hardly ramified mod $p$ Galois representation $\rho:\GQ\to\GL_2(\Z/p\Z)$ is \emph{potentially automorphic}, becoming automorphic over a (possibly non-solvable) totally real Galois extension $F/\Q$ which is disjoint from the subfield cut out by the kernel of $\rho$.
-%\end{theorem}
-% Thm 5.3 Taylor notes
-%\begin{proof}
-%  This proof is deep. There are three major steps.
-%
-%  1) We first use Moret-Bailly's theorem to find an F and an F-point $A/F$ on a moduli space of elliptic curves with $p$-torsion $\rho$ and $\ell$-torsion induced from a certain character.
-%
-%  2) We then lift the character to characteristic zero, and prove that the resulting theta series is modular (using for example a converse theorem). We deduce that $A[\ell]$ is modular.
-%
-%  3) We now use a modularity lifting theorem (due to Wiles/Taylor/Khare-Wintenberger/Skinner-Wiles/who?) to deduce that $A$ is modular. We immediately deduce that $A[p]$ is modular as required.
-%\end{proof}
-

blueprint/src/chapter/ch04overview.tex

@@ -1,11 +1,13 @@
 \chapter{An overview of the proof}\label{ch_overview}
 
-So far we have seen that, modulo Mazur's theorem, Fermat's Last Theorem can be reduced
+So far we have seen that, modulo Mazur's theorem (and various other things which will still take some work to formalise but which are much easier), Fermat's Last Theorem can be reduced
 to the statement that there is no prime $\ell\geq 5$ and hardly-ramified
 irreducible 2-dimensional Galois representation $\rho:\GQ\to\GL_2(\Z/\ell\Z)$.
 
-In this short chapter we explain our strategy for proving this. This short chapter is written
-for experts who want an overview of the route we're taking.
+In this chapter we give an overview of our strategy for proving this, and collect
+various results which we will need along the way. Note that we no longer need to assume that $\rho$ comes from the $\ell$-torsion in an elliptic curve.
+
+\section{Potential modularity.}
 
 We will only speak about modularity for 2-dimensional representations of the
 absolute Galois group of a totally real field $F$ of even degree over $\Q$, and
@@ -24,19 +26,27 @@ \chapter{An overview of the proof}\label{ch_overview}
     local behaviour;
     \item several nontrivial results in global class field theory;
     \item the Jacquet--Langlands correspondence;
+    \item The assertion that irreducible 2-dimensional mod $p$ representations induced from a character are modular (this follows from converse theorems);
     \item A modularity lifting theorem.
 \end{itemize}
 
-Everything here is from the 20th century and standard, other than the modularity
-lifting theorem, which is explained in \href{https://math.berkeley.edu/~fengt/249A_2018.pdf}
-{Taylor's 2018 Stanford course}. The strategy is to use Moret--Bailly to find an auxiliary 
-elliptic curve over $F$ whose mod $\ell$ Galois representation is $\rho$ and whose
+Everything here is from the 20th century, and most of it is from the 1980s or before.
+The exception is the modularity lifting theorem, which we state explicitly here.
+
+**TODO** state modularity lifting theorem.
+
+I am not entirely sure where to find a proof of this in the literature, although it has certainly been known to the experts for some time. Theorem 3.3 of~\cite{taylor-mero-cont} comes close, although it assumes that $\ell$ is totally split in $F$ rather than just unramified. Another near-reference is Theorem~5.2 of~\cite{toby-modularity}, although this assumes
+the slightly stronger assumption that the image of $\rho$ contains $\SL_2(\Z/p\Z)$ (however it is well-known to the experts that this can be weakened to give the result we need). One reference for the proof is \href{https://math.berkeley.edu/~fengt/249A_2018.pdf}{Richard Taylor's 2018 Stanford course}. 
+
+Given the modularity lifting theorem, the strategy to show potential modularity of $\rho$ is to use Moret--Bailly to find an appropriate totally real field $F$, an auxilary prime $p$, and an auxiliary elliptic curve over $F$ whose mod $\ell$ Galois representation is $\rho$ and whose
 mod $p$ Galois representation is induced from a character. By converse theorems (for example)
 the mod $p$ Galois representation is associated to an automorphic representation of
 $\GL_2/F$ and hence by Jacquet--Langlands it is modular. Now we use the
-modularity lifting theorem to deduce the modularity of the curve and hence
+modularity lifting theorem to deduce the modularity of the curve over $F$ and hence
 the modularity of the $\ell$-torsion. 
 
+\section{Compatible families, and reduction at 3}
+
 We now use Khare--Wintenberger to lift $\rho$ to a potentially modular $\ell$-adic
 Galois representation of conductor 2, and put it into an $\ell$-adic famiily using
 the Brauer's theorem trick in \cite{blggt}. Finally we look at the 3-adic specialisation
@@ -46,7 +56,11 @@ \chapter{An overview of the proof}\label{ch_overview}
 One can now go on to deduce that the 3-adic representation must be reducible, which
 contradicts the irreducibility of $\rho$.
 
-As this document grows, we will be able to add links to a more detailed discussion of
-what is going on here.
+As this document grows, we will add a far more detailed discussion of
+what is going on here. 
+
+
+
+
 
 

blueprint/src/chapter/ch05bestiary.tex

@@ -0,0 +1,23 @@
+\chapter{A collection of results which are needed in the proof.}
+
+**TODO** write this properly.
+
+Mention Mazur's theorem.
+
+Then: 
+
+JL, 
+
+mult 1, 
+
+Galois rep associated to an auto rep, 
+
+Definition of an automorphic representation for the units of a quaternion algebra over a totally real field (including situations where the algebra is split at one or two infinite places).
+
+Shimura curves and Shimura surfaces, plus a description of their etale cohomology in terms of automorphic representations.
+
+Classification of finite subgroups of PGL_2(F_p-bar)
+
+Moret-Bailly
+
+mor to come