  We use theta series and modular forms to prove that Z^n is the only integral
unimodular lattice of rank n without characteristic vectors of norm <n, i.e.
the only integral unimodular lattice not containing a vector w such that
(w,w)<n and 2|(v,v+w) for all lattice vectors v. By the work of Kronheimer and
others on the Seiberg-Witten equation this yields an alternative proof of a
theorem of Donaldson on the geometry of 4-manifolds.
