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<p>Approximation theorems for algebraic stacks over a number field <i>k</i> are studied in this article. For <i>G</i> a connected linear algebraic group over a number field we prove strong approximation with Brauer-Manin obstruction for the classifying stack <i>BG</i>. This result answers a very concrete question, given <i>G</i>-torsors <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P_{v}\)</EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k_{v}\)</EquationSource> </InlineEquation>, where <i>v</i> ranges over a finite number of places, when can you approximate the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(P_{v}\)</EquationSource> </InlineEquation> by a <i>G</i>-torsor <i>P</i> defined over <i>k</i>.</p>

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Approximation Theorems for Classifying Stacks over Number Fields

  • 期刊论文

  • 发表日期：

    2026年2月27日

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  • Ajneet Dhillon

摘要

Approximation theorems for algebraic stacks over a number field k are studied in this article. For G a connected linear algebraic group over a number field we prove strong approximation with Brauer-Manin obstruction for the classifying stack BG. This result answers a very concrete question, given G-torsors \(P_{v}\) over \(k_{v}\) , where v ranges over a finite number of places, when can you approximate the \(P_{v}\) by a G-torsor P defined over k.