关于旋量亏格的亏格对应
Genus-correspondences respecting spinor genus
摘要：
对于两个正定整数三元二次型$f$和$g$以及一个正整数$n$，如果$n\cdot g$用$f$和$n\cdot dg=df$表示，则称这对$(f,g)$是可按比例缩放$n$表示的对. $\text{gen}(f)\times \text{gen}(g)$中所有可表示对的集合称为属对应。Jagy指出，如果$n$是平方自由的，并且$f$的亏格中旋量亏格的个数等于$g$的亏格中旋量亏格的个数，那么这样的亏格对应在某种意义上尊重旋量亏格，即对于任何通过缩放$n$，$f' \in \text{spn}(f)$表示的对$(f,g), (f',g')$当且仅当$g' \in \text{spn}(g)$。在这篇文章中，我们通过给出一个反例来证明Jagy猜想是不成立的。此外，我们还给出了亏格对应尊重旋量亏格的一个充要条件。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
