Spinor norms of rotations of local integral quadratic forms

Abstract

Let ( F, p) be a quadratic ramified extension of the field Q 2 of 2-adic numbers, with D its ring of integers and u the group of units of D. Let L be a regular n-ary quadratic lattice over D with n ≥ 3 and sL ⊆ D. A lower bound is obtained for ord p dL when u ⊈ θ( O + ( L)). This yields a sufficient condition for the class number of an indefinite quadratic form over the ring of integers of a quadratic number field to be a divisor of the class number of the field.