more refactoring of Frey stuff

blueprint/src/chapter/ch03frey.tex

@@ -9,18 +9,158 @@ \section{The arithmetic of elliptic curves}
 We give an overview of the results we need, citing the literature for proofs. Everything here is standard, and most
 of it dates back to the 1970s or before.
 
-\begin{theorem}\label{Elliptic_curve_p_torsion_2d} Let $n$ be a positive integer, let $K$ be a separably closed
+\begin{theorem}\label{Elliptic_curve_p_torsion_size}\lean{EllipticCurve.n_torsion_card}\tangled\discussion{12345}
+  Let $n$ be a positive integer, let $K$ be a separably closed
   field with $n$ nonzero in $K$, and let $E$ be an elliptic curve over $K$. Then the $n$-torsion $E(K)[n]$ 
-  in the $K$-points of $E$ is a finite group isomorphic to $(\Z/n\Z)^2$.
+  in the $K$-points of $E$ is a finite group of size $n^2$.
 \end{theorem}
 \begin{proof}
   There are several proofs in the textbooks. The proof we shall formalise is forthcoming work of David Angdinata; it follows the approach with division polynomials, and it will be part of his PhD thesis.
 \end{proof}
 
+This theorem actually tells us the structure of the $n$-torsion, because of the following
+purely group-theoretic result:
+\begin{lemma}\label{group_theory_lemma}
+  If $n$ is a positive integer and $A$ is a finite
+  abelian group of size $n^2$ and if for all $d\mid n$, the $d$-torsion in $A$ has size $d^2$, 
+  then $A\cong (\Z/n\Z)^2$. 
+\end{lemma}
+\begin{proof}
+  One proof (which doesn't sound like much fun to formalise) writes $A$ as $\prod_{i=1}^t(\Z/a_i\Z)$
+  with $a_i\mid a_{i+1}$, uses $d=a_1$ to deduce $t=2$ and then uses $d=a_2$ to deduce $a_1=a_2$.
+\end{proof}
+
+\begin{corollary}\label{Elliptic_curve_p_torsion_2d}\lean{EllipticCurve.n_torsion_dimension}
+  Let $n$ be a positive integer, let $K$ be a separably closed
+  field with $n$ nonzero in $K$, and let $E$ be an elliptic curve over $K$. Then the $n$-torsion $E(K)[n]$ 
+  in the $K$-points of $E$ is a finite group isomorphic to $(\Z/n\Z)^2$.
+\end{theorem}
+\begin{proof}
+  This follows from the previous group-theoretic lemma~\ref{group_theory_lemma} and
+  theorem~\ref{Elliptic_curve_p_torsion_size}.
+\end{proof}
+
+If $K$ is now any field, and $E/K$ is an elliptic curve, then for $L$ and $M$ two fields
+which are $K$-algebras, and for $\phi:L\to M$ a $K$-algebra morphism, there is
+an induced $\Z/n\Z$-module morphism $E(L)[n]\to E(M)[n]$, which is functorial (that is,
+the identity map $L\to L$ induces the identity map on $E(L)[n]$, and composing $K$-algebra
+morphisms and then applying the construction is the same as applying the construction
+and then composing). From this it is a purely formal fact that for $L/K$ a Galois extension,
+$\Gal(L/K)$ acts naturally on $E(L)[n]$. If $K=\Q$, $L=\overline{\Q}$, $n>0$ and $E$ is an elliptic
+curve over $\Q$, this gives us a \emph{Galois representation} $\Gal(\Qbar/\Q)\to\GL_2(\Z/n\Z)$
+coming from the $n$-torsion of $E(\Qbar)$.
+
+A fundamental fact about this Galois representation is that its determinant is the
+cyclotomic character.
+
+\begin{theorem}\label{Elliptic_curve_det_p_torsion}\uses{Elliptic_curve_p_torsion_2d} If $E$ is an 
+  elliptic curve over a field $K$, and $n>0$ is a positive integer which is nonzero in $K$, then the 
+  determinant of the 2-dimensional representation of $\Gal(K^\sep/K)$ on $E[n]$ is the 
+  mod $n$ cyclotomic character.
+\end{theorem}
+\begin{proof}
+  This presumably should be done via the Weil pairing. I have not yet put any thought into a feasible way to go about this.
+\end{proof}
+
+\section{Good reduction}
+
+If $E$ is now an elliptic curve over the field of fractions $K$ of a DVR $R$ with maximal ideal 
+$\m$, then we say $E$ has \emph{good reduction} if $E$ has a model with coefficients in $R$ and
+the reduction mod $\m$ is still non-singular. If $E$ is an elliptic curve over a number field $N$
+then one says that $E$ has good reduction at a finite place $P$ of $N$ if the base extension of $E$
+to the completion $N_P$ of $N$ at $P$ has good reduction.
+
+\begin{example} If $E$ is the Frey curve $Y^2=X(X-a^p)(X+b^p)$ and $q$ is a prime
+  not dividing $abc$ (and in particular $q>2$), then the reduction mod $q$ of this 
+  equation is still a smooth
+  plane cubic, because the three roots $0$, $a^p$ and $-b^p$ are distinct modulo $q$
+  (note that the difference between $a^p$ and $-b^p$ is $c^p$). Hence the Frey curve
+  has good reduction at $q$.
+\end{example}
+
+If $E$ is an elliptic curve over a number field $N$ and if $\rho$ is the representation
+of $\Gal(\overline{N}/N)$ on the $n$-torsion of $E$ then the image of $\rho$ is finite,
+so by the fundamental theorem of Galois theory the representation factors through an
+injection $\Gal(L/N)\to\GL_2(\Z/n\Z)$ where $L/N$ is a finite Galois extension of
+number fields. One says that $\rho$ is \emph{unramified} at a finite place $P$ of $N$
+if $L/N$ is unramified at $P$.
+
+At some point we will need a theory of finite flat group schemes over an affine base. Here
+is a working definition.
+
+\begin{definition}\label{finite_flat_group_scheme} If $R$ is a commutative ring, then
+  a \emph{finite flat group scheme} over $R$ is the spectrum of a commutative Hopf algebra $H/R$ 
+  which is finite and flat as an $R$-module.
+\end{definition}
+
+Some facts we will need are:
+
+\begin{theorem}\label{good_reduction_implies_unramified} If $E$ is an ellipitic curve over a number 
+  field $N$ and $E$ has good reduction at a finite place $P$ of $N$, and if furthermore 
+  $n\not\in P$, then the Galois representation on the $n$-torsion of $E$ is unramified.
+\end{theorem}
+\begin{proof}
+  One approach to this would be by developing the theory of finite flat group schemes
+  and considering the $n$-torsion on a good integral model for $E$. I have not really thought
+  through whether this is the best approach. 
+\end{proof}
 
+\begin{theorem}\label{good_reduction_implies_flat} If $E$ is an ellipitic curve over a number field 
+  $N$ and $E$ has good reduction at a finite place $P$ of $N$ containing the prime number $p$, 
+  then the Galois representation on the $p$-torsion of $E$ comes from a finite flat group scheme 
+  over the integers of the completion $N_P$.
+\end{theorem}
+\begin{proof}
+  Indeed, the kernel of the $p$-torsion on a good integral model is finite and flat.
+  Checking this claim will involve a fair amount of work.
+\end{proof}
 
+\section{Multiplicative reduction}
+
+If $E$ is an elliptic curve over the field of fractions $K$ of a DVR $R$ with maximal ideal $\m$,
+then we say that $E$ has multiplicative reduction if there is an integral model of $E$
+which reduces mod $R/\m$ to a plane cubic with only one singularity, which is an ordinary double point.
+We say that the reduction is \emph{split} if the two tangent lines at the ordinary double point
+are both defined over $R/\m$, and \emph{non-split} otherwise.
+
+\begin{example} If $E$ is the Frey curve $Y^2=X(X-a^p)(X+b^p)$ and $q$ is an odd prime
+  which does divide $abc$ then the reduction mod $q$ of this equation is smooth
+  away from the point $(x,0)$ where $x$ is the double root of the cubic mod $q$,
+  and has an ordinary double point at $(x,0)$. Hence the Frey curve has
+  multiplicative reduction at $q$.
+\end{example}
+
+\begin{example} If $E$ is the Frey curve $Y^2=X(X-a^p)(X+b^p)$ associated to a Frey package
+  then $E$ has multiplicative reduction at 2. For the change of variables $X=4X'$ and $Y=8Y'+4X'$
+  changes the equation to $64Y'^2+64X'Y'=64X'^3+16X'^2(b^p-a^p-1)-4X'a^pb^p$ and, because $p\geq5$,
+  $b$ is even and $a=3$ mod 4 the 64s cancel, given an equation which reduces mod 2 to
+  $Y'^2+X'Y'=X'^3+cX'^2$ for some $c\in\{0,1\}$, a cubic which is smooth away from an ordinary 
+  double point at $(0,0)$. Hence the Frey curve has multiplicative reduction at~2. Note
+  that $c=0$ iff $E$ has split multiplicative reduction (which happens iff $a^p=7$ mod $8$).
+\end{example}
+
+In particular, the Frey curve is \emph{semistable} -- it has good or multiplicative
+reduction at all primes.
+
+The main thing we need about elliptic curves with multiplicative reduction over $p$-adic fields
+is the uniformisation theorem, originally due to Tate. 
+
+\begin{theorem}\label{Tate_curve_uniformisation} If $E$ is an elliptic curve over a field
+  complete with respect to a nontrivial nonarchimedean (real-valued) norm $K$ amd if $E$ has split
+  multiplicative reduction, then there is a Galois-equivariant injection
+  $\overline{K}^\times/q^{\mathbb{Z}}\to E(\overline{K})$.
+\end{theorem}
+\begin{proof}
+  See~\cite{silverman2}, Theorems V.3.1, Remark V.3.1.2 (we don't need surjectivity),
+  and Theorem V.5.3.
+\end{proof}
 
+\begin{corollary}\label{multiplicative_reduction_torsion} If $E$ is an elliptic curve
+  over a field $K$ complete with respet to a nontrivial nonarchimedean (real-valued) norm
+  and with perfect residue field, and if $E$ has multiplicative reduction, then the $n$-torsion
+  $E(K^\sep)[n]$ 
 
+******************************THE BELOW IS UNSORTED IDEAS****
 
 Let $\rho:\GQ\to\GL_2(\Z/p\Z)$ be the representation on the $p$-torsion of this curve. In this chapter we discuss some basic properties of this representation, used both by Mazur to prove that $\rho$ cannot be reducible and by Wiles to prove that it can't be irreducible.
 
@@ -60,11 +200,7 @@ \section{Hardly ramified representations}
 The theorem will follow from the results below. The first three are valid for all elliptic curves, the rest are specific to Frey curves.
 
 
-\begin{theorem}\label{Elliptic_curve_det_p_torsion}\uses{Elliptic_curve_p_torsion_2d} If $E$ is an elliptic curve over a field $K$ of characteristic zero, and $p$ is a prime, then the determinant of the 2-dimensional Galois representation $E[p]$ is the mod $p$ cyclotomic character.
-\end{theorem}
-\begin{proof}
-  This presumably should be done via the Weil pairing. I have not yet put any thought into a feasible way to go about this.
-\end{proof}
+
 
 \begin{theorem}\label{Elliptic_curve_quotient_by_finite_subgroup} If $E$ is an elliptic curve over a field $K$ of characteristic zero, and $p$ is a prime, and if $C\subseteq E[p]$ is a Galois-stable
   subgroup of order $p$, then there's an elliptic curve ``''$E/C$'' over $K$ and an isogeny of elliptic curves $E\to E/C$ with kernel precisely $K$.
@@ -93,10 +229,7 @@ \section{Hardly ramified representations}
 
 For primes dividing $abc$ we will need a theory of the Tate curve.
 
-\begin{definition}\label{Tate_curve_uniformisation} We will need the Tate uniformisation of an elliptic curve with
-  multiplicative reduction. A good reference for this is Silverman's second book on elliptic curves. {\bf TODO
-  } concrete reference, get biblio working.
-\end{definition}
+
 
 For primes $\ell\not\in\{2,p\}$ which do divide $abc$, the Frey curve has bad (multiplicative) reduction at $\ell$. However the miracle is that the $p$-torsion of the Frey curve does is unramified anyway.
 
@@ -114,10 +247,7 @@ \section{Hardly ramified representations}
   Again this follows from the theory of Tate uniformisation, and the fact that the Frey curve has multiplicative reduction at 2.
 \end{proof}
 
-\begin{definition}\label{finite_flat_group_scheme} We will need a definition of finite flat group schemes over a base ring. I propose using Hopf algebras.
-\end{definition}
-
 \begin{theorem}\label{Frey_curve_mod_p_rep_at_p} Let $\rho$ be the $p$-torsion in a $p$-Frey curve. Then the restriction of $\rho$ to $\GQp$ comes from a finite flat group scheme.
 \end{theorem}
 \begin{proof} The Frey curve either has good reduction at $p$ (case 1 of FLT) or multiplicative reduction at $p$ (case 2 of FLT). In the first case the $p$-torsion is unramified at $p$ by general theory, and in the second case the theory of the Tate curve shows that the $p$-torsion is (up to quadratic twist) an \emph{unramified} extension of the trivial character by the cyclotomic character.
-\end{proof}
+\end{proof}