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On almost strong approximation for linear algebraic groups

Yang Cao, Yijin Wang

TL;DR

The paper addresses almost strong approximation for connected linear algebraic groups over number fields by reducing ASA to a finiteness condition on a Brauer-Manin obstruction group B_S(G). It develops abelian Galois cohomology for reductive groups via a two-term torus complex, and establishes a precise criterion: ASA off S holds if and only if B_S(G) is finite, with the index bounded by |B_S(G)|. Leveraging Dirichlet density and the splitting field data, the authors generalize previous results (Rapinchuk–Tralle) and extend ASA to inner forms under suitable splitting hypotheses. They also bound the ASA index in terms of density and cohomology, yielding explicit results for tori and classical groups such as GL_n and PGL_n, thereby clarifying the obstruction-theoretic underpinnings of ASA in global fields.

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.

On almost strong approximation for linear algebraic groups

TL;DR

The paper addresses almost strong approximation for connected linear algebraic groups over number fields by reducing ASA to a finiteness condition on a Brauer-Manin obstruction group B_S(G). It develops abelian Galois cohomology for reductive groups via a two-term torus complex, and establishes a precise criterion: ASA off S holds if and only if B_S(G) is finite, with the index bounded by |B_S(G)|. Leveraging Dirichlet density and the splitting field data, the authors generalize previous results (Rapinchuk–Tralle) and extend ASA to inner forms under suitable splitting hypotheses. They also bound the ASA index in terms of density and cohomology, yielding explicit results for tori and classical groups such as GL_n and PGL_n, thereby clarifying the obstruction-theoretic underpinnings of ASA in global fields.

Abstract

Let be a connected linear algebraic group over a number field . In this article, we study the almost strong approximation property (ASA) of 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 . Using the criteria, we conclude that (ASA) can be completely controlled by the Dirichlet density of the places and the splitting field of , which generalizes a result of Rapinchuk and Tralle.

Paper Structure

This paper contains 8 sections, 28 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Almost strong approximation and Dirichlet density
  3. Almost strong approximation and the Brauer-Manin obstruction
  4. Notations, terminology and preliminary results
  5. Abelian Galois cohomology of reductive groups
  6. The proof of the Theorem \ref{['thm-mainlem']}
  7. The proof of Theorem \ref{['thm-mainthm']} and Theorem \ref{['thm-(ASA) with inner forms']}
  8. The index of almost strong approximation

Key Result

Theorem 1.4

Let $K$ be a number field and $G$ a semi-simple simply connected algebraic group over $K$. Let $S$ be a finite non-empty set of places such that $G'_{S}:=\prod_{v\in S}G'(K_{v})$ is non-compact for any almost $K$-simple factor $G'$ of $G$. Then $G$ satisfies (SA) off $S$.

Theorems & Definitions (60)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.4: Kneser, Platonov
  • Definition 1.5: AW25
  • Example 1.6: Ra12, Proposition 2.1
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.12: De11,Corollary 3.20
  • ...and 50 more