Classification of one-class spinor genera for quaternary quadratic forms

TL;DR

This paper classifies primitive quaternary quadratic forms based on their spinor genus properties, identifying the unique case where the spinor genus is a singleton but the genus is not.

Contribution

It provides a complete classification of primitive quaternary quadratic forms with a one-class spinor genus that is not a one-class genus.

Findings

Only one primitive quaternary genus has a one-class spinor genus but not a one-class genus.

Most primitive quaternary genera have coinciding genus and spinor genus.

In other cases, genera split into multiple spinor genera and classes.

Abstract

A quadratic form has a one-class spinor genus if its spinor genus consists of a single equivalence class. In this paper, we determine that there is only one primitive quaternary genus which has a one-class spinor genus but not a one-class genus. In all other cases, the genera of primitive quaternary lattices either have a genus and spinor genus which coincide, or the the genus splits into multiple spinor genera, which in turn split into multiple equivalence classes.