Completeness of the list of spinor regular ternary quadratic forms

TL;DR

This paper proves that the known list of primitive spinor regular ternary quadratic forms is complete, confirming no other such forms exist beyond the 29 identified by previous computational searches.

Contribution

The paper establishes the completeness of the list of primitive spinor regular ternary quadratic forms, resolving a long-standing classification question.

Findings

No additional primitive spinor regular ternary quadratic forms exist beyond the known 29.

The classification confirms the finiteness of such forms.

The result completes the understanding of spinor regularity in ternary quadratic forms.

Abstract

Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.