Trivialité des groupes de Whitehead réduits avec applications à l’approximation faible et l’approximation forte

Yong Hu; Yisheng Tian

doi:10.5802/crmath.705

Research article - Number theory

Trivialité des groupes de Whitehead réduits avec applications à l’approximation faible et l’approximation forte
[Triviality of reduced Whitehead groups with applications to weak approximation and strong approximation]

Yong Hu ¹; Yisheng Tian ²

¹ Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
² Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 465-477.

Résumé (Original language)
Abstract

Nous prouvons certains résultats de trivialité pour des groupes de Whitehead réduits et groupes de Whitehead unitaires réduits pour des algèbres à division sur un corps de valuation discrète hensélien dont le corps résiduel a dimension cohomologique virtuelle ou séparable $\le 2$. Ces résultats sont appliqués pour démontrer l’approximation forte pour des groupes simplement connexes absolument presque simples isotropes de type A. Comme cas particulier, un tel groupe défini sur le corps des fonctions d’une courbe non réelle $C/k$ vérifie l’approximation forte si le corps de base $k$ est un corps de nombres, un corps $p$-adique, $\mathbb{C}(\!(t)\!)$ ou un corps de fonctions à deux variables sur $\mathbb{R}$.

We prove some triviality results for reduced Whitehead groups and reduced unitary Whitehead groups for division algebras over a Henselian discrete valuation field whose residue field has virtual cohomological dimension or separable dimension $\le 2$. These results are applied to show strong approximation for isotropic absolutely almost simple simply connected groups of type A. In particular, such a group defined over the function field of a nonreal curve $C/k$ satisfies strong approximation if the base field $k$ is a number field, a $p$-adic field, $\mathbb{C}(\!(t)\!)$ or a two-variable function field over $\mathbb{R}$.

Received: 2023-01-10
Revised: 2024-01-10
Accepted: 2024-11-18
Published online: 2025-05-26

DOI: 10.5802/crmath.705

Classification: 11E57, 19B99, 20G35, 16K20
Mots-clés : Groupe de Whitehead réduit, groupe de Whitehead unitaire, algèbre à division sur un corps hensélien, approximation faible, approximation forte, groupes simplement connexes
Keywords: Reduced Whitehead group, unitary Whitehead group, division algebra over a Henselian field, weak approximation, strong approximation, simply connected groups

Author's affiliations:

Yong Hu ¹; Yisheng Tian ²

¹ Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
² Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Yong Hu and Yisheng Tian},
     title = {Trivialit\'e des groupes de {Whitehead} r\'eduits avec applications \`a l{\textquoteright}approximation faible et l{\textquoteright}approximation forte},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {465--477},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2025},
     doi = {10.5802/crmath.705},
     language = {fr},
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Yong Hu; Yisheng Tian. Trivialité des groupes de Whitehead réduits avec applications à l’approximation faible et l’approximation forte. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 465-477. doi : 10.5802/crmath.705. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.705/

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