有限域上阿贝尔簇的Frobenius Alexander纽胚、纽着色与同构类
Frobenius Alexander quandles, knot colorings, and isogeny classes of abelian varieties over finite fields
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中文总结 AI 辅助
该研究引入有限域上阿贝尔簇的Frobenius Alexander纽胚概念，确定纽被其非平凡着色的素数条件，证明其同构类可确定阿贝尔簇同构类，为纽理论与算术研究提供新工具。
中文摘要 AI 辅助
我们引入与有限域上阿贝尔簇相关的Frobenius Alexander纽胚的概念，并从纽理论与算术两个视角对其展开研究。我们依据纽的Alexander多项式与阿贝尔簇的Frobenius特征多项式的结式，精确确定了纽在哪些有理素数下可被这些纽胚赋予非平凡着色。这引出了持久可着色性的概念及其刻画，以及在纤维化纽和小亏格纽上的应用。我们还证明了一个刚性结果，表明Frobenius Alexander纽胚的同构类决定了对应挠子群上Frobenius自同态的相似类。由此可得，一个足够大的Frobenius Alexander纽胚可确定Frobenius自同态的特征多项式，进而确定给定有限基域上阿贝尔簇的同构类。
英文摘要
We introduce the notion of Frobenius Alexander quandles associated with abelian varieties over finite fields, and study them from both knot-theoretic and arithmetic viewpoints. We determine exactly for which rational primes a knot admits a nonconstant coloring by these quandles, in terms of the resultant of its Alexander polynomial and the Frobenius characteristic polynomial of the abelian variety. This leads to the notion of persistent colorability and its characterization, as well as applications to fibered knots and knots of small genus. We also prove a rigidity result showing that the isomorphism class of a Frobenius Alexander quandle determines the similarity class of the Frobenius endomorphism on the corresponding torsion subgroup. As a consequence, one sufficiently large Frobenius Alexander quandle determines the characteristic polynomial of the Frobenius endomorphism, and hence, determines the isogeny class of the abelian variety over the given finite base field.
