Finite quadratic modules and lattices

TL;DR

This paper provides a new proof that finite quadratic modules can be decomposed into indecomposables and constructs minimal-rank lattices with prescribed discriminant modules, explicitly given by Gram matrices.

Contribution

It introduces a novel proof of the decomposition theorem and constructs minimal-rank lattices corresponding to indecomposable modules with explicit Gram matrices.

Findings

Finite quadratic modules can be decomposed into indecomposables.

Constructed minimal-rank lattices with given discriminant modules.

Explicit Gram matrices for the lattices are provided.

Abstract

We give a new proof of the fact that any finite quadratic module can be decomposed into indecomposable ones. For any indecomposable finite quadratic module, we construct a lattice, and a positive definite lattice, both of which are of the least rank, whose discriminant module is the given one. The resulting lattices are given by their Gram matrices explicitly.