Linear algebraic groups with good reduction
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RESEARCH
Linear algebraic groups with good reduction Andrei S. Rapinchuk1 and Igor A. Rapinchuk2* * Correspondence:
[email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA Full list of author information is available at the end of the article 2
Abstract 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.
1 Introduction Over the past six decades, the analysis of various properties of linear algebraic groups over local and global fields, the origins of which can be traced to the works of Lagrange and Gauss, has developed into a well-established theory known as the arithmetic theory of algebraic groups (cf. [94]). While this subject remains an area of active research, there is growing interest in the arithmetic properties of linear algebraic groups over fields of an arithmetic nature that are not global (such as function fields of curves over various classes of fields, including p-adic fields and number fields). These recent developments rely on a symbiosis of methods from the theory of algebraic groups on the one hand, and arithmetic geometry on the other. At this stage, it is too early to give a comprehensive account of these new trends, so the goal of the present article is to discuss one important, and somewhat surprising, instance of the propagation of the ideas of arithmetic geometry into the theory of algebraic groups. Curiously, reduction techniques that have been used in the analysis of diophantine equations since antiquity, and the notion of good reduction, which is central to modern arithmetic geometry, were utilized in the classical arithmetic theory of algebraic groups in a rather limited way (see the discussion in Sect. 5).
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A. S. Rapinchuk, I. A. Rapinchuk Res Math Sci (2020)7:28
A key novelty in the current work is that the consideration of algebraic groups having good reduction at an appropriate set of discrete valuations of the base field has moved to the forefront. In fact, one of the important conjectures in the area states that
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