Direct-sum cancellation of lattices over orders in global fields

We study the cancellation question for lattices (finitely generated torsion-free modules) over orders in algebraic number fields: Given lattices L, M and N with L ⊕ M ≅ L ⊕ N, when can one conclude that M ≅ N? Some definitive results in the quadratic case were obtained about twenty years ago. Here we concentrate on the case of cubic and higher-degree number fields, where very different techniques are needed. The cubic case appears to be quite difficult, and our results in this case are very incomplete. Perhaps surprisingly, number fields of degree four or more are more tractable, and we have a definitive answer to the cancellation question for a large family of orders in these fields. Our results apply also to the case of algebraic function fields in one variable over a finite field of constants.