论文标题
在有理函数的后轨道中的环形细胞学和阿贝尔点
Cyclotomic and abelian points in backward orbits of rational functions
论文作者
论文摘要
我们证明了在数字字段上有理函数的向后轨道上证明了几个结果。首先，我们表明，如果$ k $是一个数字字段，则在k（x）$中$ ϕ \ in k $中的$ ϕ \，则仅在有限的很多prime上，Abelian点在$α$的后轨道中生成的$ K $的扩展。这有很大的后果是，如果$α$的落后轨道中的所有点都是阿贝利安，那么$ ϕ $都是急剧有限的。 We use this result to prove two facts: on the one hand, if $ϕ\in \mathbb Q(x)$ is a quadratic rational function not conjugate over $\mathbb Q^{\text{ab}}$ to a power or a Chebyshev map and all preimages of $α$ are abelian, we show that $ϕ$ is $\mathbb Q$-conjugate to one在最近的猜想安德鲁斯和Petsche的精神的精神上。另一方面，我们为$ k（x）$中的二次合理函数提供条件，以$ k（x）$的向后轨道$α$的向后轨道，以仅包含有限的许多环体元素预图，从而扩大了第二作者的先前结果。最后，我们为三重$（ϕ，k，α）$提供了必要的条件，其中$ ϕ $是数字字段$ k $和$α\ in k $的lattès映射，整个$α$的$α\ in K $仅包含Abelian点。
We prove several results on backward orbits of rational functions over number fields. First, we show that if $K$ is a number field, $ϕ\in K(x)$ and $α\in K$ then the extension of $K$ generated by the abelian points in the backward orbit of $α$ is ramified only at finitely many primes. This has the immediate strong consequence that if all points in the backward orbit of $α$ are abelian then $ϕ$ is post-critically finite. We use this result to prove two facts: on the one hand, if $ϕ\in \mathbb Q(x)$ is a quadratic rational function not conjugate over $\mathbb Q^{\text{ab}}$ to a power or a Chebyshev map and all preimages of $α$ are abelian, we show that $ϕ$ is $\mathbb Q$-conjugate to one of two specific quadratic functions, in the spirit of a recent conjecture of Andrews and Petsche. On the other hand we provide conditions on a quadratic rational function in $K(x)$ for the backward orbit of a point $α$ to only contain finitely many cyclotomic preimages, extending previous results of the second author. Finally, we give necessary and sufficient conditions for a triple $(ϕ,K,α)$, where $ϕ$ is a Lattès map over a number field $K$ and $α\in K$ for the whole backward orbit of $α$ to only contain abelian points.
