$\Kab$ 对典范高度具有 Bogomolov 性质

中文摘要 AI 辅助

我们证明，对于任何数域 $K$ 和 $K(x)$ 中不与幂、Chebyshev 或 Lattès 映射共轭的任何有理映射 $f$，$K^{\mathrm{ab}}$ 对 $f$ 的典范高度具有强 Bogomolov 性质。我们还分类了后向轨道包含无穷多个阿贝尔点的对 $(f,\alpha)$。这解决了 Andrews--Petsche 猜想对数域上的有理映射以及后向轨道的无穷阿贝尔子集的情形。作者在与 \\(\emph{Astra}\\) 的对话中得到了证明的主要思想。关键见解在于应用等分布结果 \cite{Yua08}，随后对 $(f,f)$-预周期曲线进行分类 \cite{Pak23,Pak20, Bea25}，再在参数化的预周期曲线上额外应用等分布。

英文摘要

We show that for any number field $K$ and any rational map $f$ in $K(x)$ that is not conjugate to a power, (signed) Chebyshev or Lattès map, then $K^{\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. We also classify the pairs $(f,α)$ whose backward orbit contains infinitely many abelian points. This settles the Andrews--Petsche conjecture to rational maps over a number field and to infinite abelian subsets of backward orbits. The authors were led to the main idea of the proof in conversation with \emph{Astra}. The key insight consists of applying the equidistribution results \cite{Yua08}, followed by a classification of $(f,f)$-preperiodic curves \cite{Pak23,Pak20, Bea25} followed by an additional application of equidistribution on a parametrized preperiodic curve.

发表机构

• Google（谷歌）

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