论文标题
统一的Artin型问题ii
Unified treatment of Artin-type problems II
论文作者
论文摘要
这项工作涉及Artin对数字字段原始根和相关问题的猜想。让$ k $为一个数字字段，让$ w_1 $ to $ w_n $是有限生成的子组，为$ k^\ times $ as usation等级。我们考虑索引图，该索引图将$ \ Mathfrak P $ $ k $映射到$ n $ -tuple $（w_i \ bmod \ mathfrak p）$的$ n $ tuple。有条件地，在GRH下，指数图下的任何前图都承认了密度，而这项工作的目的正在描述它。例如，我们以各种方式将密度表示为限制。我们特别研究了通过规定其条目的估值来定义的$ n $ tuplass集的预先图。在一些轻度的假设下，我们可以将密度表示为（适当定义）Artin型常数的倍数。
This work concerns Artin's Conjecture on primitive roots and related problems for number fields. Let $K$ be a number field and let $W_1$ to $W_n$ be finitely generated subgroups of $K^\times$ of positive rank. We consider the index map, which maps a prime $\mathfrak p$ of $K$ to the $n$-tuple of the indices of $(W_i \bmod \mathfrak p)$. Conditionally under GRH, any preimage under the index map admits a density, and the aim of this work is describing it. For example, we express the density as a limit in various ways. We study in particular the preimages of sets of $n$-tuples that are defined by prescribing valuations for their entries. Under some mild assumptions we can express the density as a multiple of a (suitably defined) Artin-type constant.
