This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but now it appears to be developing into one of the central topics in the emerging arithmetic theory of (linear) algebraic groups over higher-dimensional fields. The focus of this article is on the Main Conjecture (Conjecture 5.7) asserting the finiteness of the number of isomorphism classes of forms of a given reductive group over a finitely generated field that have good reduction at a divisorial set of places of the field. Various connections between this conjecture and other problems in the theory of algebraic groups (such as the analysis of the global-to-local map in Galois cohomology and the genus problem) are discussed in detail. The article also includes a brief review of the required facts about discrete valuations, forms of algebraic groups, and Galois cohomology.

中文翻译：

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线性代数群具有良好的约简

本文是对还原性代数群的猜想和结果的调查，这些代数群在适当的基础场离散估值集下具有良好的归约性。直到最近，这个问题还很少受到关注，但是现在，它似乎正在发展成为新兴的高维场（线性）代数组算术理论的中心主题之一。本文的重点是主要猜想（猜想5.7），它断言了在有限生成的场上给定的还原基团形式的同构类数的数量的有限性，这些场在场的除数集上具有良好的归约性。详细讨论了这个猜想与代数群论中的其他问题之间的各种联系（例如，在伽罗瓦同调学中全局到局部图的分析和类问题）。本文还简要回顾了有关离散估值，代数形式和Galois同调性的必要事实。