On almost strong approximation for linear algebraic groups

TL;DR

This paper investigates the almost strong approximation property for connected linear algebraic groups over number fields, providing a criterion based on the Brauer group and relating it to the group's splitting field and places.

Contribution

It introduces a necessary and sufficient condition for (ASA) using the Brauer group, extending previous results by linking (ASA) to Dirichlet density and splitting fields.

Findings

(ASA) can be characterized by the Brauer group of G.

The (ASA) property is controlled by Dirichlet density and the splitting field.

The results generalize previous work by Rapinchuk and Tralle.

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.