Dual systems of integral vectors (general questions and applications to the geometry of positive quadratic forms)

After giving a brief introduction to our new “theory of dual systems of integer vectors”, we give the first applications to the theory of positive quadratic forms. We consider the question of enumerating the $L$-polytopes of lattices, paying particular attention to the case of five-dimensional lattices. The results reported in this paper were announced earlier by the authors in Doklady [1]; here we give the details.