Class numbers of ternary quadratic forms

TL;DR

This paper investigates how Watson transformations affect the class numbers of positive definite ternary quadratic forms, offering a new method for computing these class numbers and deriving explicit formulas for various genera.

Contribution

It introduces a novel approach to compute class numbers of ternary quadratic forms using Watson transformations, with explicit formulas for multiple genera.

Findings

Watson transformations preserve key arithmetic properties.

The method simplifies class number calculations for certain genera.

Explicit class number formulas are derived for many cases.

Abstract

G.L. Watson \cite{watson1, watson2} introduced a set of transformations, called Watson transformations by most recent authors, in his study of the arithmetic of integral quadratic forms. These transformations change an integral quadratic form to another integral quadratic form with a smaller discriminants, but preserve many arithmetic properties at the same time. In this paper, we study the change of class numbers of positive definite ternary integral quadratic formula along a sequence of Watson transformations, thus providing a new and effective way to compute the class number of positive definite ternary integral quadratic forms. Explicit class number formulae for many genera of positive definite ternary integral quadratic forms are derived as illustrations of our method.