Some upper bounds on dense module sublattices from number fields

In this paper, we consider the problem of finding dense rank one module sublattices in module lattices of a given rank, where by module lattice we mean a lattice obtained as [Formula: see text]-linear combinations of vectors with coefficients in a number field [Formula: see text], with ring of integers [Formula: see text], and [Formula: see text] an order. We provide generic upper bounds on the normalized volume of rank one module sublattices for [Formula: see text] totally real or CM, specific bounds for quadratic extensions and some cyclotomic extensions, as well as exact values for some norm Euclidean quadratic extensions.