The geometry of numbers over algebraic number fields

Abstract

1. The Geometry of Numbers was founded by Minkowski in order to attack certain arithmetical problems, and is normally concerned with lattices over the rational integers. Minkowski himself, however, also treated a special problem over complex quadratic number fields [5], and a number of writers have since followed him. They were largely concerned with those fields which have class-number h= 1; and this simplification removes many of the characteristic features of the more general case. Hermann Weyl [10] gave a thorough account of the extension of Minkowski's theory of the reduction of quadratic forms to gauge functions over general algebraic number fields and quaternion algebras, and we shall follow part of his developments, though our definition of a lattice is quite different. The desirability of extending the Geometry of Numbers to general algebraic number fields was emphasized by Mahler in a seminar at Princeton. In this paper we shall carry out this program, extending the fundamental results of Mahler [4] to our more general case and applying them to specific problems. Certain new ideas are necessary, but much of this paper must be regarded as expository. In particular, when the proof of a result is essentially analogous to that for the real case we have merely given a reference to the latter. 2. Let K be an algebraic extension of the rationals of degree m. We regard K as an algebra over the rationals, which we can extend to an algebra K* over the reals. It is well known that K* is commutative and semi-simple (being in fact isomorphic to the direct sum of r copies of the reals and s copies of the complex numbers, where r and 2s are the number of real and complex conjugates of K); and the integers of K* are just those of K. We now define the n-dimensional space Kn over K as being the set of ordered n-tuples of elements in K*. Any tCK* is of the form t=x1i1+ * * * +xwmU, where the x, are real and co,, * * *, com is an integral basis for K; and hence there is a natural map of Kn onto Rmn in which each component t is mapped onto m of the components of the point in Rmn, namely xi, x, m as above. We can define a metric and a measure in Kn by means of those in Rmn, with the above map, and so Kn is a locally compact complete metric space.