On the existence of free sublattices of bounded index and arithmetic applications

Let $\mathcal{O}$ be a Dedekind domain whose field of fractions $K$ is a global field. Let $A$ be a finite-dimensional separable $K$-algebra and let $\mathrm{\Lambda }$ be an $\mathcal{O}$-order in $A$. Let $n$ be a positive integer and suppose that $X$ is a $\mathrm{\Lambda }$-lattice such that $K{\otimes }_{\mathcal{O}}X$ is free of rank $n$ over $A$. Then $X$ contains a (non-unique) free $\mathrm{\Lambda }$-sublattice of rank $n$. The main result of the present article is to show there exists such a sublattice $Y$ such that the generalised module index $[X:Y{]}_{\mathcal{O}}$ has explicit upper bounds with respect to division that are independent of $X$ and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties.