Heights of subspaces over noncommutative fields. (Q870635)

This paper generalizes the notion of height of subspaces \(V\) in \(K^N\) where \(K\) is a number field and \(N\) is a natural number to the case of \(D^N\) where \(D\) is a finite-dimensional rational division algebra. The generalization is based on a definition of height in terms of Euclidean lattices found in \textit{W.\ Schmidt} [Ann. Math. (2) 85, 430--472 (1967; Zbl 0152.03602)]. Matrix versions of this are also considered. The authors develop a number of analogous results based on this notion of height, such as a version of Siegel's Lemma over \(D\), and then give a formula for the degree of an abelian subvariety \(Y\) of a power of a simple abelian variety \(X\) with endomorphism algebra \(D\).