Genus-correspondences respecting spinor genus

TL;DR

This paper investigates the properties of genus-correspondences between positive definite ternary quadratic forms, disproves a conjecture by Jagy regarding spinor genus respect, and establishes a precise condition for when such correspondences respect spinor genus.

Contribution

The paper provides a counterexample to Jagy's conjecture and derives a necessary and sufficient condition for genus-correspondences to respect spinor genus.

Findings

Counterexample disproves Jagy's conjecture.

Necessary and sufficient condition established.

Clarifies the relationship between genus-correspondences and spinor genus.

Abstract

For two positive definite integral ternary quadratic forms $f$ and $g$ and a positive integer $n$, if $n\cdot g$ is represented by $f$ and $n\cdot dg=df$, then the pair $(f,g)$ is called a representable pair by scaling $n$. The set of all representable pairs in $\text{gen}(f)\times \text{gen}(g)$ is called a genus-correspondence. Jagy conjectured that if $n$ is square free and the number of spinor genera in the genus of $f$ equals to the number of spinor genera in the genus of $g$, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs $(f,g), (f',g')$ by scaling $n$, $f' \in \text{spn}(f)$ if and only if $g' \in \text{spn}(g)$. In this article, we show that by giving a counter example, Jagy's conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.