Watson transformations and class numbers of ternary quadratic forms

Abstract

The class number of an integral quadratic form is defined by the number of inequivalent classes in its genus. Recently, Chan and Oh gave an explicit relation between the class number of a (positive definite integral) ternary quadratic form and the class number of its Watson transformation at an odd prime. In this thesis, we consider the case when the Watson transformation is taken at the prime 2 or 4 on arbitrary ternary quadratic forms and give an explicit relation under this situation. We finally give an effective inductive method on the computation of the class number of an arbitrary ternary quadratic form. As an example, we provide the class number formula for any Bell ternary quadratic form.