AI 中文总结
本文针对整体函数域𝔽_q(t)的A₄四次扩域，引入带装饰的Hurwitz空间参数化哈塞范数原理的失效，结合切博塔廖夫密度定理证明其失效通常极为罕见。
AI 中文摘要
对于整体域的有限扩域K/k，范数映射N_{K/k}: K^× → k^×可延拓为伊代尔群上的映射。若k^×中每个在所有局部均为范数的元素也都是整体范数，则称哈塞范数原理成立。本文研究数域语境下难以触及的情形——A₄四次扩域中哈塞范数原理的统计规律，证明对于整体函数域𝔽_q(t)的A₄四次扩域，哈塞范数原理的失效通常极为罕见。为此，本文引入一种带装饰的Hurwitz空间来参数化哈塞范数原理的失效情形，再利用切博塔廖夫密度定理估计其出现频率。
英文摘要
For a finite extension of global fields $K/k$, the norm map $N_{K/k} : K^\times \to k^\times$ extends to a map on idèle groups. The Hasse norm principle holds if every element of $k^\times$ that is a norm everywhere locally is also a norm globally. In this paper, we study the statistics of the Hasse norm principle in a setting that is out of reach in the number field context, namely that of $A_4$-quartic extensions. We show that failures of the Hasse norm principle are generally rare for $A_4$-quartic extensions of global function fields $\mathbb{F}_q(t)$. We achieve this by introducing a decorated Hurwitz space parametrising the failures of the Hasse norm principle and then using the Chebotarev density theorem to estimate their frequency.
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