论文标题
二次形式和属理论：与2-depent的链接，并应用于理想阶级的非平凡专业
Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes
论文作者
论文摘要
属理论是整体二元二元形式的经典特征。利用作者对二次形式类和二次代数的理想类之间的众所周知对应关系的概括，我们将其扩展到扭曲二次形式并在任何PID $ r $中具有系数的情况。当$ {r = \ mathbb {k} [x]} $时，我们表明属理论映射是一定的超ellip曲曲线上$ 2 $ deScent地图的二次形式。作为应用程序，我们对Agboola和Pappas的问题做出了贡献，内容涉及在高纤维曲线上的Divisor类专业问题。在适当的假设下，我们证明了一组非平凡的专业化的密度$ 1 $。
Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any PID $R$. When ${R = \mathbb{K}[X]}$, we show that the Genus Theory map is the quadratic form version of the $2$-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of non-trivial specializations has density $1$.
