The Integral Extension of Isometries of Quadratic Forms Over Local Fields

Abstract

Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.