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arXiv 2202.01673math.NTmath.DS
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有理映射下轨道的$p$ -Adic插值

$p$-Adic interpolation of orbits under rational maps

Jason P. Bell, Xiao Zhong

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摘要：

设$L$是一个特征为零的域，$h:\mathbb{P}^1\to \mathbb{P}^1$是定义在$L$上的有理映射，$c\in \mathbb{P}^1(L)$ .我们发现，存在一个生成的子字段$K$的$L$上的$c$和$h$的定义，以及一个无限集的不等价的非阿基米德完成$K_{\mathfrak{p}}$，其中存在一个正整数$a=a(\mathfrak{p})$的属性，$i\in \{0,\ldots ,a-1\}$存在一个幂级数$g_i(t)\in K_{\mathfrak{p}}[[t]]$，收敛于$K_{\mathfrak{p}}$的封闭单位盘，使$h^{an+i}(c)=g_i(n)$的所有足够大的$n$。结果证明了动力学Mordell-Lang猜想对$\mathbb{P}^1 \times X$的分裂自映射$(h,g)$和$g$成立.

英文摘要：

Let $L$ be a field of characteristic zero, let $h:\mathbb{P}^1\to \mathbb{P}^1$ be a rational map defined over $L$, and let $c\in \mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and $h$ are defined along with an infinite set of inequivalent non-archimedean completions $K_{\mathfrak{p}}$ for which there exists a positive integer $a=a(\mathfrak{p})$ with the property that for $i\in \{0,\ldots ,a-1\}$ there exists a power series $g_i(t)\in K_{\mathfrak{p}}[[t]]$ that converges on the closed unit disc of $K_{\mathfrak{p}}$ such that $h^{an+i}(c)=g_i(n)$ for all sufficiently large $n$. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps $(h,g)$ of $\mathbb{P}^1 \times X$ with $g$ étale.

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