On the existence of free sublattices of bounded index and arithmetic applications (Q6588188)

Let \({\mathcal O}\) be a Dedekind domain whose field of fractions is a global field \(K\ne \mathcal{O}\). Let \(A\) be a separable finite-dimensional \(K\)-algebra and let \(\Lambda\) be an order in \(A\). Let \(\mathcal{M}\) be a maximal \(\mathcal{O}\)-order of \(A\) containing \(\Lambda\) and let \(J\) be a full two-sided ideal of \(\mathcal{M}\) contained in \(\Lambda\). Let \(\mathcal{K}\) be a nonzero ideal of \(\mathcal{O}\). Then there exists a nonzero ideal \(\mathcal{I}\) of \(\mathcal{O}\) that is relatively prime to \(\mathcal{K}\) such that the following holds: Given any \(\Lambda\)-lattice \(X\) such that \(KX\) is free of rank \(n\) over \(A\), there exists a free \(\Lambda\)-sublattice \(Z\) of \(X\) such that the module index \([X:Z]_{\mathcal{O}}\) divides \(\mathcal{I}\cdot [\mathcal{M} : \Lambda]_{\mathcal{O}}^{2n}\) if \(A\) is commutative or \(\mathcal{I}\cdot [\mathcal{M} : J]_{\mathcal{O}}^n \cdot [\mathcal{M} : \Lambda]_{\mathcal{O}}^{n}\) otherwise.\N\NMore precise results are given when \(\Lambda\) is the group ring \(\mathbb Z[G]\) of a finite group. Applications are given to strong Minkowski units, approximations of normal integral bases, and the Galois module structure of rational points of abelian varieties over number fields.