2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78605We 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.6 pagesNumber TheoryDifferential Geometry11H55 (Primary) 11F11, 11H06, 57R55 (Secondary)text