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\centerline{\Large
  Yet another proof of the uniqueness of the $E_8$ lattice
  }
\vspace*{3ex}
\centerline{Noam D. Elkies}
\centerline{August, 2004}
\vspace*{6ex}

{\bf Theorem} (Mordell 1938~\cite{Mordell}){\bf .}  {\em
Let $L$ be an even unimodular positive-definite lattice of rank~$8$.
Then $L \cong E_8$.
}

\vspace*{3ex}

{\em Yet another proof}\/:
We construct an odd ``$2$-neighbor'' $L'$ of~$L$,
identify $L'$ with $\Z^8$ using our characterization of~$\Z^n$
by its minimal characteristic norm~\cite{odd1},
and use this to identify $L$ with~$E_8$.

A ``$2$-neighbor'' of~$L$ is a self-dual lattice
$L' \subset L \otimes \Q$ such that $L_0 = L \cap L'$
is a sublattice of index~$2$ in each of~$L$ and~$L'$.
To obtain $L_0$ and~$L'$,
first fix a lattice vector $v_0 \notin 2L$ such that $4 | (v_0,v_0)$.
(Such a vector exists in every even lattice of rank at least~$3$.)
We may assume that $(v_0,v_0) \equiv 4 \bmod 8$, since if
$8 | (v_0,v_0)$ we may replace $v_0$ by $v_0+2w$ with $(v_0,w)$ odd.
Set
$$
L_0 = \{ v \in L : 2 | (v,v_0) \},
\qquad
L' = L_0 \cup (L_0 + \frac{v_0}{2}).
$$
We claim that $L'$ is an odd self-dual lattice.
That it is a lattice is clear since certainly $v_0 \in L_0$.
It is contained in its dual by construction.
Since $[L':L_0] = [L:L_0] = 2$, we have $\disc(L')=\disc(L)=1$,
so $L' \subseteq {L'}^*$ implies $L' = {L'}^*$.
Finally $L'$ is odd because it contains the vector $v_0/2$
of odd norm.

Recall (see for instance~\cite{odd1})
that every unimodular lattice~$M$\/ has characteristic vectors:
lattice vectors $c\in M$\/ such that $(c,v)\equiv(v,v)\bmod 2$
for all $v\in M$.  These constitute a coset of~$2M$\/ in~$M$,
and if $M$\/ is positive definite of rank~$n$
then $(c,c) \equiv n \bmod 8$ for all characteristic $c\in M$.
Since $L'$ is odd, $0$ is not a characteristic vector of~$L'$.
Hence for every characteristic vector~$c$ of~$L'$ we have
$(c,c) \geq 8 = \rank(L')$.
By~\cite{odd1} it follows that $L' \cong \Z^8$.

Therefore $L_0$, being an even lattice
contained with index~$2$ in~$L'$, is a $D_8$ lattice.
Thus $L$ is one of the three lattices
intermediate between $D_8$ and $D_8^*$.  One of these lattices
is $L' = \Z^8$.  The other two are isomorphic to~$E_8$, Q.E.D.

\vspace*{3ex}

{\bf Acknowledgements.} 
This work was supported in part by the
the National Science Foundation (grant DMS-0200687),
and took place during a workshop at
the American Institute of Mathematics.

\begin{thebibliography}{CS1}
\bibitem{odd1} Elkies, N.D.:
A characterization of the $\Z^n$ lattice.
{\em Math.\ Res.\ Letters} {\bf 2} \#3 (5/1995), 321--326. \
{\tt math.NT/9906019} at {\tt arXiv.org}.
\bibitem{Mordell} Mordell, L.J.:
The definite quadratic forms in eight variables with determinant unity,
{\em J. Math.\ Pures Appl.}\ {\bf 17} (1938), 41--46.
\end{thebibliography}

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