Representations by unimodular quadratic \(\mathbb{Z}\)-lattices (Q1323412)

The author's recent results on primitive representations of lattices by unimodular lattices in Am. J. Math. 113, 129-146 (1991; Zbl 0729.11016), J. Number Theory 44, 356-366 (1993; Zbl 0781.11017) are extended. Necessary and sufficient local conditions are given for the primitive representation of a lattice by a unimodular quadratic lattice \(L\) over the integers of a local field (with 2 not ramified), and the number of inequivalent representations, under the action of the orthogonal group O\((L)\), determined. Global results over \(\mathbb{Z}\), for forms corresponding to the Dynkin diagrams, are then obtained via strong approximation.