On the construction of indecomposable positive definite quadratic forms over \({\mathbb Z}\)
Precise results on the existence of indecomposable positive definite lattices over \({\mathbb{Z}}\) with given rank and given discriminant were given by \textit{O. T. O'Meara} [J. Reine Angew. Math. 276, 99--123 (1975; Zbl 0304.10013)] for dimension \(n\geq 10\). (A lattice \(L\) is called decomposable, if it can be expressed as the orthogonal sum of two non-zero sublattices: \(L=M\perp N.)\) This paper extends these results to \(3\leq n\leq 9\), where O'Meara left some cases unsettled. For \(n=2\) a certain criterion for the existence of indecomposable lattices is given by analytic methods.
indecomposable positive definite quadratic forms
scientific article; zbMATH DE number 4075176