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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 6-8: Lattices
September 10, 12 and 14, 2001
\end{center}

\head{Reminder} If you didn't submit a questionnaire, please at least
give me your email address (preferably by sending me an email at
\verb+kedlaya@math+). I'll use email and the course web site for 
some announcements.

\head{Reference}
Neukirch, Sections 1.4 and 1.5.

\head{Jargon watch}
Warning: when I say ``lattice'', I usually mean what the book calls a
``complete lattice''. If not, I'll say ``partial lattice''.
Also, I will use the more common term \emph{fundamental domain} for what 
the book
calls a \emph{fundamental mesh}, that is, a subset of a real vector space
whose translates by some lattice exactly cover the vector space.

A \emph{real embedding} of a number field $K$ is an injection $K \hookrightarrow \RR$. A \emph{complex embedding} is an injection $K \hookrightarrow \CC$
whose image is not contained in $\RR$. Complex embeddings come in pairs
that are interchanged by applying complex conjugation to $\CC$; the
\emph{signature} of $K$ is the pair $(r,s)$, where $r$ is the number of real
embeddings and $s$ is the number of pairs of complex embeddings.

\head{Outline of lectures}

\begin{enumerate}
\item
Define a (complete) lattice in several different ways:
\begin{enumerate}
\item[(a)] as a subgroup of a real vector space generated by a basis;
\item[(b)] as a discrete subgroup of a real vector space;
\item[(c)] as a free abelian group with a positive definite inner product;
\item[(d)] as a free abelian group with a positive norm.
\end{enumerate}
\item
Define the volume of a lattice. Note that passing to a sublattice
of index $n$ multiplies the volume by $n$.
\item
State Minkowski's lattice theorem: any centrally symmetric, convex subset
of $\RR^n$ of volume greater than $2^n$ contains a nonzero element of any
lattice of volume 1.
\item
Deduce from Minkowski's theorem that a rank $n$ lattice of volume $V$
contains a nonzero vector of norm at most $2 (V/B_n)^{1/n}$, where
$B_n$ is the volume of the $n$-ball.
\item
Define real embeddings, complex embeddings, and
the signature of a number field.
\item
Given a number field, define the Minkowski space (as the product of the
real and complex embeddings).
\item
Reinterpret the trace pairing as the Minkowski metric, and
the discriminant of a basis
as its Minkowski volume.
\item
Define the multiplicative Minkowski space (to be used later).
\end{enumerate}

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