AI 中文总结
本文利用Bockstein谱序列与李理论工具改进了Labute的方法，验证了无穷多3位集合对应的数域pro-p伽罗瓦群满足一致Fontaine-Mazur性质，还构造了此类无限群并给出高概率检测的数值证据。
AI 中文摘要
对于数域$K$、素数$p$以及有限的tame位集合$S$，我们研究群$G_{K,S}$——即$K$的在$S$外非分歧的极大pro-$p$扩张的伽罗瓦群。Tame Fontaine–Mazur猜想预测，这些群不存在非平凡的一致幂pro-$p$商群。\n 本文中，我们利用Bockstein谱序列与李理论工具，提出了一种解决该问题的新方法。这使得我们能够推广并改进J. Labute此前提出的一种方法，Labute仅能处理对每个$\u27f8\u27e9\notin S$都满足$p^2\nmid N(\u27f8\u27e9)-1$的情形。\n 基于此，我们提出了一种方法，可验证大量$|S|=3$的任意$G_{K,S}$的一致Fontaine–Mazur性质。在$K$满足温和条件的情况下，我们证明存在无穷多三元组$S$，使得对应的群$G_{K,S}$没有非平凡的一致商群。我们还构造了一大类此类无限群。最后，我们给出的数值证据表明，对于$|S|=3$的情形，本文提出的判别准则能以极高概率检测出一致Fontaine–Mazur性质。
英文摘要
For a number field $K$, a prime $p$ and a finite set of tame places $S$ we consider the groups $G_{K,S}$ - the Galois groups of the maximal pro-$p$ extension of $K$ unramified outside $S$. The tame Fontaine--Mazur Conjecture predicts that these groups have no nontrivial uniformly powerful pro-$p$ quotients. In this paper we develop a new approach to this problem using Bockstein spectral sequences and Lie-theoretic tools. This allows us to extend and refine an earlier method due to J.~Labute, who was only able to consider the case where $p^2\nmid N(\mathfrak{q})-1$ for each $\mathfrak{q}\in S$. Based on this we develop a method to verify the uniform Fontaine--Mazur property for many $G_{K,S}$ with $|S|=3$ arbitrary. Under mild conditions on $K$ we show that for infinitely many triples $S$, the groups $G_{K,S}$ have no nontrivial uniform quotients. We also exhibit a large class of these groups, which are infinite. Finally, we present numerical evidence indicating that the criteria developed here detect the uniform Fontaine--Mazur property with very high probability for $|S|=3$.
Comments30 pages; Comments are welcome!
