The Hasse principle for lattices in a full matrix algebra

Abstract

There naturally arises the question of similarity of these lattices, i.e., whether there exists, for given lattices K and L, an element g~GL (n,F) such that g-ILg = K. We consider this question from the point of view of the Hasse principle, which, as will become clear, does not hold in this situation. Therefore, we introduce a measure of deviation (the number of genus classes) and obtain some estimates. As a corollary we obtain a result on the class number, in the sense of [i], for the projective linear group. The following definitions were inspired by the classical work of Kneser [2, 3].