\classheader{2013-01-30}

There is still a bit more to say about number theory before we get back to algebra.

On June 23, 1993, Wiles attended a conference in England where he gave three talks. There were rumors, so people knew that something was going to be special, and on the third day he announced his proof. I was one of the organizers of the conference, but I was in Japan at the time.

There were t-shirts made for the occasion,
\begin{center}
\begin{tikzpicture}
\draw[thick] (-1,-1) rectangle (1,2);
\draw[thick] (1,1) rectangle (1.5,1.5);
\draw[thick] (-1,1) rectangle (-1.5,1.5);
\node at (0,0) {\small In 1637\ldots};
\draw[thick] (-0.4,0.5) rectangle (0.4,1);
\draw[thick] (-0.4,1) to (0,1.2) to (0.4,1);
\end{tikzpicture}
\end{center}

I've received several letters where people who were not professionals claimed to have proven Fermat's Last Theorem. They were all making trivial manipulations until they made a mistake, and thought they had reached a contradiction.

There is a myth of Lorelei, a maiden who lived on a rock in the Rhine river who would distract fishermen with her song so they forgot to control their boat, and they ended up on the bottom of the river. Many people spent their lives in vain trying to prove Fermat's Last Theorem.

Last time, I didn't mention what happens in the number field situation for elliptic curves, only what happens modulo $p$.

An elliptic curve $E/\Q$ is a curve of the form $y^2=f(x)$ where $f(x)\in\Q[x]$ is of degree 3, and has no multiple roots.

The set $E(K)=\{(x,y)\in K\times K\mid y^2=f(x)\}\cup\{\pointatinfinity\}$ is a group under the operation $+$ defined by setting $P+Q+R=\pointatinfinity$ when $P$, $Q$, and $R$ are collinear:

\begin{center}
\begin{tikzpicture}
\begin{axis}[no markers, axis lines=none,axis y line=center,axis equal, xmin=-3.5, xmax=3.5, ymin=-3.5, ymax=3.5,width=7cm, height=7cm]
\addplot +[no markers,raw gnuplot,thick] gnuplot {
set contour base;
set cntrparam levels discrete 0.00;
unset surface;
set view map;
set isosamples 1000;
set xrange[-4:3];
set yrange[-3:3];
splot y^2-x^3-1;
};
\end{axis}
\draw[thick,red] (1,1.5) to (5,5.5);
\node[mypoint,label=left:$P$] at (1.95,2.45) {};
\node[mypoint,label=below right:$Q$] at (2.98,3.48) {};
\node[mypoint,label=left:$R$] at (3.95,4.45) {};
\end{tikzpicture}
\end{center}

(special cases are needed when the line is tangent to the curve, or is vertical.)

On $E:y^2=x^3+1$, we have that $E[6]$ (the 6-torsion of $E$) is isomorphic to $\Z/6\Z$, consisting of $\{\pointatinfinity,P,2P,3P,4P,5P\}$ where $P=(2,3)$:

\begin{center}
\begin{center}
\begin{tikzpicture}
\begin{axis}[title={$y^2=x^3+1$},no markers, axis lines=none,axis y line=center,axis equal, xmin=-3.5, xmax=3.5, ymin=-3.5, ymax=3.5]
\addplot +[no markers,raw gnuplot,thick] gnuplot {
set contour base;
set cntrparam levels discrete 0.00;
unset surface;
set view map;
set isosamples 1000;
set xrange[-3.5:3.5];
set yrange[-3.5:3.5];
splot y^2-x^3-1;
};
\addplot [only marks] table {
2  3
0  1
-1  0
0  -1
2  -3
};
\node [below right] at (axis cs:  2,  3) {$P$};
\node [above] at (axis cs:  0,  1) {$2P$};
\node [left ] at (axis cs: -1, 0) {$3P$};
\node [below] at (axis cs:  0,  -1) {$4P$};
\node [right ] at (axis cs: 2,  -3) {$5P$};
\end{axis}
\end{tikzpicture}
\end{center}
\end{center}
Viewed on the projective plane, you can think of this as simply being
\begin{center}
\begin{tikzpicture}
\draw[thick,blue] (0,0) circle (1);
\node[mypoint,label=right:$\pointatinfinity$] at (0:1) {};
\node[mypoint,label=above right:$P$] at (60:1) {};
\node[mypoint,label=above left:$2P$] at (120:1) {};
\node[mypoint,label=left:$3P$] at (180:1) {};
\node[mypoint,label=below left:$4P$] at (240:1) {};
\node[mypoint,label=below right:$5P$] at (300:1) {};
\end{tikzpicture}
\end{center}

For a natural number $n$, we define
\[E[n]:=\{(x,y)\in E(\overline{\Q})\mid n\cdot(x,y)=\pointatinfinity\}\cong(\Z/n\Z)^2.\]
This has a natural action of $\Gal(\overline{\Q}/\Q)$ on it. There is then a number field $\Q(E[n])$, where we adjoin the $x$- and $y$-coordinates of each of the points in $E[n]$ to $\Q$.

\begin{center}
\begin{tikzcd}[column sep=0.1in]
\Gal(\overline{\Q}/\Q) \ar{rr} \ar{rd} & &\GL(2,\Z/n\Z)\\
&\Gal(\Q(E[n])/\Q) \ar[hook]{ru}
\end{tikzcd}
\end{center}

Let $p$ be an odd prime number.

Frey's elliptic curve takes a hypothetical counterexample to Fermat's Last Theorem, $a^b+b^p=c^p$, and is defined by $y^2=x(x-a^p)(x-b^p)$. Note that this is of the form
\[(x-A)(x-B)(x-C)\]
where $A-B$, $B-C$, and $A-C$ are all $p$th powers. This implies that the ramification in $\Q(E[p])/\Q$ is very, very small.

The Taniyama-Shimura conjecture gives a correspondence between elliptic curves and modular forms,
\[E\;\;\longleftrightarrow\;\; f=\sum_{n=1}^\infty a_nq^n\]
where $a_p=1+p-\# E(\F_p)$ for almost all $p$. This is a special case of the Langlands correspondence. Wiles proved a large part of this conjecture.

The property that $A-B$, $B-C$, and $C-A$ are all $p$th powers implies that $f\bmod p$ has level $\leq 2$, but the correspondence in the Taniyama-Shimura conjecture implies that such an $f$ does not exist. This establishes Fermat's Last Theorem.

The correspondence between $E$'s and $f$'s is something like the quadratic reciprocity law. The correspondence between $y^2=x^3+\cdots$ and a modular form $f$ is like the correspondence between $x^2=-1$ and $\chi:(\Z/4\Z)^\times\to\{\pm 1\}$ with $\chi(1)=1$, $\chi(3)=-1$.


Two comments:
\begin{itemize}
\item[1.] $\chi:(\Z/n\Z)^\times \to\C^\times$ is a modular form of $\GL(1,\Q)$, $f=\eta(6z)\eta(18z)$ is a modular form of $\GL(2,\Q)$, etc.
\item[2.] The Kronecker-Weber theorem says that any finite abelian extension $K$ of $\Q$ is contained in some cyclotomic field $\Q(\zeta_n)$.
\begin{center}
\begin{tikzcd}
\Gal(\Q(\zeta_n)/\Q) \ar{r}{\cong} \ar{d}[swap]{\text{restriction}} & (\Z/n\Z)^\times\ar{d}{\chi}\\
\Gal(\Q(\sqrt{m})/\Q) \ar{r}[swap]{\cong} & \{\pm 1\}
\end{tikzcd}
\end{center}
Let $p$ be a prime, and assume $p\nmid n$. Then $p$ decomposes in $\ringofintegersof{K}$ completely if and only if $p\bmod N\in H\subset(\Z/n\Z)^\times$, where $H$ is the subgroup of $(\Z/n\Z)^\times$ corresponding to $K$ in the Galois theory correspondence.
\end{itemize}

Hilbert had a rough idea of class field theory at the end of the 19th century. Takagi was very lucky to be able to study under him, not many people in Japan were able to go to Europe at the time. When he came back, the people of Takagi's village threw a festival, because a great person had returned.

Takagi's thesis under Hilbert established the following result. If $K/\Q(i)$ is a finite abelian extension, then $K$ is contained in a field of the form $\Q(i)(E[n])$ where $E:y^2=x^3-x$. Note that $(x,y)\mapsto (-x,iy)$ is an automorphism of $E$ corresponding to multiplication by $i$. This is a special case of Kronecker's dream-of-youth (Jugendtraum).

Hilbert conjectured that for any number field $K$, there was a special number field $L$ containing $K$ such that $\frak{p}\subset\ringofintegersof{K}$ decomposes completely in $\ringofintegersof{L}$ if and only if $\frak{p}$ is principal. For example, if $K=\Q(\sqrt{-5})$, then $L=\Q(\sqrt{-5},i)$. In this extension, any maximal ideal is unramified, $\frak{p}$ splits if and only if $\frak{p}$ is principal, and $\frak{p}$ remains prime if and only if $\frak{p}$ is not principal.