乘法群$\boldsymbol{\rightarrow}$的希尔伯特不可约性

中文摘要 AI 辅助

设$K$为数域，$S$为有限个非阿基米德位的集合，记$\boldsymbol{\rightarrow}$为$K$的$S$-整数环，$\boldsymbol{\rightarrow}^\times$为其单位群。设$\boldsymbol{\rightarrow}: X \rightarrow \boldsymbol{\rightarrow}^1$为定义在$K$上的（不可约）曲线的态射，记$\boldsymbol{\rightarrow}(\boldsymbol{\rightarrow})$中满足纤维$\boldsymbol{\rightarrow}^{-1}(\boldsymbol{\rightarrow})$可约（即伽罗瓦群在该纤维上的作用非传递）的元素$\boldsymbol{\rightarrow}$构成的集合为$\text{Red}(\boldsymbol{\rightarrow})$。希尔伯特不可约定理断言，$\text{Red}(\boldsymbol{\rightarrow})$包含于$\boldsymbol{\rightarrow}^1(\boldsymbol{\rightarrow})$的一个薄子集内。本文中，我们给出$\boldsymbol{\rightarrow}^\times \bigcap \text{Red}(\boldsymbol{\rightarrow})$的显式描述。作为应用，我们证明了受波利亚与西格尔经典定理启发得到的如下结果：设$p_1,\boldsymbol{\rightarrow},p_s$为有理素数，$f \boldsymbol{\rightarrow} \boldsymbol{\rightarrow}[x]$，则以下两个条件等价：1. 存在无穷多组$(e_1,\boldsymbol{\rightarrow},e_s) \boldsymbol{\rightarrow} \boldsymbol{\rightarrow}^s$使得多项式$f(x)-p_1^{e_1}\boldsymbol{\rightarrow}p_s^{e_s}$可约；2. $f = p_1^{a_1}\boldsymbol{\rightarrow}p_s^{a_s}g^\boldsymbol{\rightarrow}$（其中$\boldsymbol{\rightarrow}$为素数）或$f = -4p_1^{a_1}\boldsymbol{\rightarrow}p_s^{a_s}g^4$，其中$g \boldsymbol{\rightarrow} \boldsymbol{\rightarrow}[x]$，$a_1,\boldsymbol{\rightarrow},a_s$为整数。

英文摘要

Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $π: X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(π)$ the set of $α\in \mathbb{P}^1(K)$ such that the fibre $π^{-1}(α)$ is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that $\operatorname{Red}(π)$ is contained in a thin subset of $\mathbb{P}^1(K)$. In this paper we give an explicit description of $\mathcal{O}_S^\times \cap \operatorname{Red}(π)$. As an application we prove the following result inspired by a classical theorem of Pólya and Siegel. Let $p_1,\dotsc,p_s$ be rational primes. Let $f \in \mathbb{Q}[x]$.Then the following are equivalent: - There are infinitely many tuples $(e_1,\dotsc,e_s) \in \mathbb{N}^s$ such that the polynomial $f(x)-p_1^{e_1} \cdots p_s^{e_s}$ is reducible. - $f=p_1^{a_1} \cdots p_s^{a_s} g^\ell$ (with $\ell$ prime) or $f=-4 p_1^{a_1} \cdots p_s^{a_s} g^4$ for some $g \in \mathbb{Q}[x]$ and some integers $a_1,\dotsc,a_s$.

发表机构

• University of Warwick（华威大学）
• Universität Bayreuth（拜罗伊特大学）

机构由 AI 辅助整理，请以论文原文为准。