On almost strong approximation for linear algebraic groups

2025/11/02 by Yang Cao, Cao, Yang, Wang, Yijin
Mathematics · #Tensor decomposition and applications #Finite Group Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2511.00824

Abstract

Let G be a connected linear algebraic group over a number field K. In this article, we study the almost strong approximation property (ASA) of G raised by Rapinchuk and Tralle. Building on Demarche's results on strong approximation with Brauer-Manin obstruction, we introduce a necessary and sufficient condition for (ASA) to hold in terms of the Brauer group of G. Using the criteria, we conclude that (ASA) can be completely controlled by the Dirichlet density of the places and the splitting field of G, which generalizes a result of Rapinchuk and Tralle.

Citations

• On Almost Strong Approximation in Reductive Algebraic Groups
• Comparing descent obstruction and Brauer-Manin obstruction for open varieties
• On strong approximation for algebraic groups
• Local-global principles for 1-motives
• Arithmetic Duality Theorems for 1-Motives

Related

• On Almost Strong Approximation in Reductive Algebraic Groups
• On strong approximation for algebraic groups
• The étale Brauer-Manin obstruction to strong approximation on homogeneous spaces
• Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields
• Approximation theorems for classifying stacks over number fields