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\begin{document}
\centerline{Math 361 lecture for Wednesday, Week 10}
\bigskip

\centerline{Minkowski's theorem for lattices}
\bigskip

\textbf{Goal.} Let~$K$ be a number field.
Let $\mathcal{F}$ be the multiplicative group of nonzero fractional ideals.  A
fractional ideal is called \emph{principal} if it is generated as
an~$\mathfrak{O}_{K}$-module by a single element.  So a principal fractional
ideal has the from~$c^{-1}(\alpha)$ where $c\in K\setminus\{0\}$ and $\alpha\in
\mathfrak{O}_{K}$.  Let~$\mathcal{P}\subseteq\mathcal{F}$ denote the subgroup
of nonzero principal fractional ideals.  The \emph{class group} of~$K$ is the
quotient group
\[
  \mathcal{H} =\mathcal{F}/\mathcal{P}.
\]
The \emph{class} number $h_{K}:=h(\mathfrak{O}_{K}):=|\mathcal{H}|$, the order
of the class group.  We will see that~$h_{K}=1$ if and only if
$\mathfrak{O}_{K}$ is a PID.  In general, $h_{K}$ is a measure of how far
way~$\mathfrak{O}_{K}$ is from being a~PID.  \textbf{Our goal} is to prove that the
class number is finite.
\medskip

\textbf{Lattices in~$\R^{n}$}.

\begin{defn}
  A subset~$L\subset\R^{n}$ is a \emph{rank~$m$ lattice in $\R^{n}$} if
  $L=\Span_{\Z}\{v_{1},\ldots,v_{m}\}$ for some set $\{v_{1},\ldots,v_{m}\}$ of
  linearly independent vectors in~$\R^{n}$.
\end{defn}

A subset of~$\R^{n}$ is \emph{discrete} if its intersection with each compact
subset of~$\R^{n}$ is finite.  Equivalently, the subset has no accumulation
points.

\begin{thm}
  An additive subgroup~$L\subset\R^{n}$ is a lattice if and only if it is
  discrete.
\end{thm}

\begin{defn}
 A \emph{fundamental domain} for a rank~$n$ lattice~$L$ in $\R^{n}$ is a set of
 the form
 \[
   F=\left\{\textstyle\sum_{i=1}^{n} a_i v_i: 0\leq a_{i}<1 \text{ for $i=1,\ldots
   n$}\right\}.
 \]
 where~$L = \Span_{\Z}\{v_{1},\ldots,v_{n}\}$.
\end{defn}

\begin{remark}  With notation as in the above definition,
  \begin{enumerate}
    \item The volume of~$F$ is $\vol(F)=|\det(v_{1},\ldots,v_{n})|$.
    \item For each~$x\in\R^{n}$, there exists a unique~$\ell\in L$ such that~$x\in
      \ell+F$.
  \end{enumerate}
\end{remark}

\begin{eg}\
  \begin{enumerate}
    \item $\Z$ is a lattice in~$\R$.  Let~$S^{1}=\{z\in\C:|z|\}$ be the unit circle in $\R^{2}$
      centered at the origin.  We have a homeomorphism
      \begin{align*}
	\R/\Z\to S^{1}\\
	x\to e^{2\pi ix}.
      \end{align*}
      A fundamental domain is $[0,1)$.
    \item Consider the number field~$K=\Q(\sqrt{2})$ with number
      ring~$\Z[1,\sqrt{2}]$.  The embeddings of $K$ into $\C$ are
      $\sigma_{1}(a+b\sqrt{2})=a+b\sqrt{2}$ and
      $\sigma_{2}(a+\sqrt{2})=a-b\sqrt{2}$.  Consider the homomorphism
      \begin{align*}
	K&\to \R^{2}\\
	x&\mapsto (\sigma_{1}(x),\sigma_{2}(x)).
      \end{align*}
      Then the image of $\Z[\sqrt{2}]$ in~$\R^{2}$ is a lattice with generators
      $(\sigma_{1}(1),\sigma_{2}(1))=(1,1)$
      and $(\sigma_{1}(\sqrt{2}),\sigma_{2}(\sqrt{2}))=(\sqrt{2},-\sqrt{2})$.  The
      fundamental domain corresponding to these generators has volume
      \[
	\left|\det
	\left(\begin{array}{cc}
	    1 & \sqrt{2} \\
	    1 & -\sqrt{2}
	\end{array}\right)\right|
	= 2\sqrt{2}.
      \]
      \textbf{Exercises:} 
      \begin{enumerate}
        \item Draw a picture of this lattice and the fundamental domain
	  specified above.
	\item Can you find a different fundamental domain?  What is its volume?
      \end{enumerate}
  \end{enumerate}
\end{eg}

\begin{defn}
 The $n$-torus is the topological space
 \[
   T^{n} = \underbrace{S^{1}\times \cdots\times S^{1}}_{\text{$n$ times}}
 \]
 with the product topology.
\end{defn}

\begin{prop}
  Let~$L$ be a rank~$m$ lattice in~$\R^{n}$ with generators
  $v_{1},\ldots,v_{m}$.  Complete~$v_{1},\ldots,v_{m}$ to a
  basis~$v_{1},\ldots,v_{n}$ for~$\R^{n}$. Then there is a homeomorphism
  \begin{align*}
    \phi\colon \R^{n}/L&\to T^{m}\times \R^{n-m}\\
    \sum_{i=1}^{n} a_i v_i&\mapsto \left(e^{2\pi i a_{1}},\ldots,e^{2\pi i
    a_{m}},a_{m+1},\ldots,a_{n}\right).
  \end{align*}
  The mapping~$\phi$ is a bijection when restricted to the fundamental
  domain~$F$.
\end{prop}
\begin{proof}
 Exercise.
\end{proof}

\begin{eg}
  Consider the lattices $L=\Span_{\Z}\{(1,0),(0,1)\}$ and
  $L'=\Span_{\Z}\{(1,0)\}$ in~$\R^{2}$.  We have that~$\R^{2}/L$ is homeomorphic
  to a torus and~$\R^{2}/L'$ is homeomorphic to a cylinder.
\end{eg}

\begin{defn}
  Let~$L\subset\R^{n}$ be a rank~$n$ lattice, and consider the mapping $\pi:
  \R^{n}\to \R^{n}/L\xrightarrow{\phi}T^{n}$, the quotient mapping followed by
  the isomorphism~$\phi$ defined above.
  The \emph{volume} of $Y\subseteq T^{n}$ is defined to be 
  \[
    \vol(Y)=\vol(\phi^{-1}(Y)\cap F)
  \]
  where~$F$ is a fundamental domain for~$L$.
\end{defn}

\begin{prop}
   Let~$X\subset\R^{n}$ be a bounded such that~$\vol(X)$ exists.  With notation
   as in the above definition, suppose that $\pi$ restricted to~$X$ is
   injective.  Then~$\vol(X)=\vol(\pi(X))$.
\end{prop}
\begin{proof}
  See Theorem~6.7 and the accompanying Figure~6.6.
\end{proof}
\bigskip

\textbf{Minkowski's theorem.}

\begin{defn}
  Let~$X\subseteq\R^{n}$.  Then~$X$ is \emph{convex} if it contains the line
  segment joining each pair of points in~$X$.  In other words, if $x,y\in X$,
  then $\lambda x+(1-\lambda)y \in X$ for $\lambda\in[0,1]$.
\end{defn}

\begin{eg} If $P=\{p_{1},\ldots,p_{k}\}\subset\R^{n}$, the smallest convex set
  containing~$P$ is 
  \[
    \mathrm{conv}(P)=\left\{\textstyle\sum_{i=1}^{k} \lambda_{i}p_{i}: \text{$\lambda_{i}\geq0$
    and $\sum_{i=1}^{k} \lambda_{i}=1$}\right\}.
  \]
  This set is called the \emph{convex hull} of~$P$.
\end{eg}

\begin{defn}
 Let~$X\subseteq\R^{n}$.  Then~$X$ is \emph{centrally symmetric about the
 origin} if $x\in X$ implies $-x\in X$ for all~$x\in X$.  
 We will use the abbreviation \emph{symmetric} to mean centrally symmetric about
 the origin in the context of Minkowski's theorem.
\end{defn}

\begin{thm}[Minkowski's theorem]
  Let $L\subset\R^{n}$ be a rank~$n$ lattice, and let~$F$ be a fundamental
  domain for~$L$. Let~$X\subset\R^{n}$ be bounded, convex, and symmetric.
  Suppose that
  \[
    \vol(X)>2^{n}\vol(F).
  \]
  Then~$X$ contains a nonzero lattice point.
\end{thm}

\begin{ex}
  Consider Minkowski's theorem for the cases:
  \begin{itemize}
    \item $L=\Z\subset\R$, and
    \item $L=\Span_{\Z}\{(1,0),(0,1)\}\subset\R^{2}$.
  \end{itemize}
\end{ex}

\begin{proof}[Proof of Minkowski's theorem]
  Consider the lattice~$2L$, whose fundamental domain has volume~$2^{n}\vol(F)$.
  If~$\vol(X)>2^{n}\vol(F)$, then we have seen
  that~$\pi\colon\R^{n}\to\R^{n}/(2L)$ is not injective when restricted to $X$.
  Thus, there exist distinct $x,y\in X$ such that~$\pi(x)=\pi(y)$.  So~$x-y\in 2L$, and
  thus
  \[
    \frac{1}{2}(x-y)\in L.
  \]
  Since~$X$ is symmetric, $-y\in X$.  Since~$X$ is convex, it follows that
  \[
    \frac{1}{2}(x-y)=\frac{1}{2}x+\frac{1}{2}(-y)\in X.
  \]
  Since~$x\neq y$, we have $(x-y)/2$ is a nonzero lattice point in~$X$.
\end{proof}
\end{document}
