AI 中文总结
本文针对数域根式扩张中的小点问题，给出确保特定集合具有Bogomolov性质的通用判据，作为应用得到Rémond猜想的新案例，论证基于p-adic李扩张的相关结果。
AI 中文摘要
我们研究代数域的根式扩张中的小点。给定有理数域$\boldsymbol{\text{Q}}$的代数扩张$\boldsymbol{\text{F}}$、有限生成子群$\boldsymbol{\text{\text{Γ}}}\boldsymbol{\text{⊆}}\boldsymbol{\text{F}}^\boldsymbol{\times}$以及有理素数$\boldsymbol{\text{p}}$，我们给出一个通用判据，确保$\boldsymbol{\text{F}}(\boldsymbol{\text{Γ}}^{\boldsymbol{p}\boldsymbol{-}\boldsymbol{\text{div}}})\boldsymbol{\text{∖}}\boldsymbol{\text{Γ}}^{\boldsymbol{\text{div}}}$具有Bogomolov性质。该问题源于Rémond的一个猜想（当$\boldsymbol{\text{F}}$为数域时提出），预测此类根式扩张在用于生成它们的群的可除包之外不包含意外的小点。作为应用，我们获得了关于由有限素数集的除法点生成的根式扩张的Rémond猜想的新案例，恢复并扩展了Amoroso与第三作者的先前结果。我们的论证基于最近关于$\boldsymbol{\text{p}}$-adic李扩张中小点的结果。
英文摘要
We study small points in radical extensions of algebraic fields. Given an algebraic extension $\mathbb{F}$ of $\mathbb{Q}$, a finitely generated subgroup $Γ\subseteq \mathbb{F}^\times$, and a rational prime $p$, we give a general criterion ensuring that $\mathbb{F}(Γ^{p-\mathrm{div}})\setminus Γ^{\mathrm{div}}$ has the Bogomolov property. This problem is motivated by a conjecture of Rémond, formulated when $\mathbb{F}$ is a number field, predicting that such radical extensions contain no unexpected small points outside the divisible hull of the group used to generate them. As applications, we obtain new cases of Rémond's conjecture for radical extensions generated by division points with respect to a finite set of primes, recovering and extending previous results of Amoroso and the third author. Our argument is based on a recent result on small points in $p$-adic Lie extensions.
Comments18 pages, comments are welcome!
