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

Article de recherche - Théorie des nombres

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

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 (VO)
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 $\leq 2$ . 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, $C ((t))$ ou un corps de fonctions à deux variables sur $R$ .

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 $\leq 2$ . 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, $C ((t))$ or a two-variable function field over $R$ .

Reçu le : 2023-01-10
Révisé le : 2024-01-10
Accepté le : 2024-11-18
Publié le : 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

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G5_465_0,
     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},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.705},
     language = {fr},
}

TY  - JOUR
AU  - Yong Hu
AU  - Yisheng Tian
TI  - Trivialité des groupes de Whitehead réduits avec applications à l’approximation faible et l’approximation forte
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 465
EP  - 477
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.705
LA  - fr
ID  - CRMATH_2025__363_G5_465_0
ER  -

%0 Journal Article
%A Yong Hu
%A Yisheng Tian
%T Trivialité des groupes de Whitehead réduits avec applications à l’approximation faible et l’approximation forte
%J Comptes Rendus. Mathématique
%D 2025
%P 465-477
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.705
%G fr
%F CRMATH_2025__363_G5_465_0

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/

Bibliographie
Cité par

[1] Eva Bayer-Fluckiger; Raman Parimala Classical groups and the Hasse principle, Ann. Math. (2), Volume 147 (1998) no. 3, pp. 651-693 | DOI | MR | Zbl

[2] Olivier Benoist The period-index problem for real surfaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 130 (2019), pp. 63-110 | DOI | MR | Zbl

[3] Vladimir Chernousov; Alexander Merkurjev $R$ -equivalence and special unitary groups, J. Algebra, Volume 209 (1998) no. 1, pp. 175-198 | DOI | MR

[4] Jean-Louis Colliot-Thélène Quelques résultats de finitude pour le groupe ${S K}_{1}$ d’une algèbre de biquaternions, $K$ -Theory, Volume 10 (1996) no. 1, pp. 31-48 | DOI | MR | Zbl

[5] Jean-Louis Colliot-Thélène; Philippe Gille; Raman Parimala Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J., Volume 121 (2004) no. 2, pp. 285-341 | DOI | MR | Zbl

[6] Peter Draxl Skew fields, London Mathematical Society Lecture Note Series, 81, Cambridge University Press, 1983, ix+182 pages | DOI | MR

[7] Ido Efrat Valuations, orderings, and Milnor $K$ -theory, Mathematical Surveys and Monographs, 124, American Mathematical Society, 2006, xiv+288 pages | DOI | MR

[8] Yurii Leonidovich Ershov Henselian valuations of division rings and the group ${S K}_{1}$ , Mat. Sb., N. Ser., Volume 117(159) (1982) no. 1, pp. 60-68 | MR

[9] Philippe Gille Invariants cohomologiques de Rost en caractéristique positive, $K$ -Theory, Volume 21 (2000) no. 1, pp. 57-100 | DOI | MR | Zbl

[10] Philippe Gille Le problème de Kneser–Tits, Séminaire Bourbaki. Volume 2007/2008. Exposés 982–996 (Astérisque), 2009 no. 326, pp. 39-81 | Numdam | MR | Zbl

[11] Philippe Gille Groupes algébriques semi-simples en dimension cohomologique $\leq$ 2, Lecture Notes in Mathematics, 2238, Springer, 2019, xxii+167 pages | DOI | MR

[12] Philippe Gille; Tamás Szamuely Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, 165, Cambridge University Press, 2017, xi+417 pages | DOI | MR

[13] Nicolas Grenier-Boley On the triviality of certain Whitehead groups, Math. Proc. R. Ir. Acad., Volume 107 (2007) no. 2, pp. 183-193 | DOI | MR | Zbl

[14] Yong Hu A cohomological Hasse principle over two-dimensional local rings, Int. Math. Res. Not. (2017) no. 14, pp. 4369-4397 | DOI | MR | Zbl

[15] Yong Hu; Jing Liu; Yisheng Tian Strong approximation and Hasse principle for integral quadratic forms over affine curves, Acta Arith., Volume 216 (2024) no. 3, pp. 277-289 | DOI | MR | Zbl

[16] Bill Jacob; Adrian R. Wadsworth Division algebras over Henselian fields, J. Algebra, Volume 128 (1990) no. 1, pp. 126-179 | DOI | MR | Zbl

[17] Max-Albert Knus; Alexander Merkurjev; Markus Rost; Jean-Pierre Tignol The book of involutions, Colloquium Publications, 44, American Mathematical Society, 1998, xxii+593 pages | DOI | MR

[18] Tsit Yuen Lam Introduction to quadratic forms over fields, Graduate Studies in Mathematics, 67, American Mathematical Society, 2005, xxii+550 pages | MR

[19] Valery Antonovich Lipnitskiĭ Norms in fields of real algebraic functions and the reduced Whitehead group, Dokl. Akad. Nauk BSSR, Volume 26 (1982) no. 7, pp. 585-588 | MR | Zbl

[20] Alexander Merkurjev Certain $K$ -cohomology groups of Severi–Brauer varieties, $K$ -theory and algebraic geometry : connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) (Proceedings of Symposia in Pure Mathematics), American Mathematical Society, 1995 no. 58, part 2, pp. 319-331 | DOI | MR | Zbl

[21] Alexander Merkurjev Invariants of algebraic groups, J. Reine Angew. Math., Volume 508 (1999), pp. 127-156 | DOI | MR | Zbl

[22] Alexander Merkurjev; Andrei Suslin $K$ -cohomology of Severi–Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 46 (1982) no. 5, pp. 1011-1046 | MR

[23] Raman Parimala; Venapally Suresh Period-index and u-invariant questions for function fields over complete discretely valued fields, Invent. Math., Volume 197 (2014) no. 1, pp. 215-235 | DOI | MR | Zbl

[24] Vladimir Platonov; Vyacheslav I. Yanchevskiĭ The structure of unitary groups and of the commutants of a simple algebra over global fields, Dokl. Akad. Nauk SSSR, Volume 208 (1973), pp. 541-544 | MR

[25] David Saltman Division algebras over $p$ -adic curves, J. Ramanujan Math. Soc., Volume 12 (1997) no. 1, pp. 25-47 | MR | Zbl

[26] Jean-Pierre Serre Local fields, Graduate Texts in Mathematics, 67, Springer, 1979, viii+241 pages | DOI | MR

[27] Jean-Pierre Serre Cohomologie galoisienne, Lecture Notes in Mathematics, 5, Springer, 1994, x+181 pages | DOI | MR

[28] Abhay Soman On triviality of the reduced Whitehead group over Henselian fields, Arch. Math., Volume 113 (2019) no. 3, pp. 237-245 | DOI | Zbl

[29] Andrei Suslin ${S K}_{1}$ of division algebras and Galois cohomology revisited, Proceedings of the St. Petersburg Mathematical Society. Vol. XII (American Mathematical Society Translations, Series 2), American Mathematical Society, 2006 no. 219, pp. 125-147 | DOI | Zbl

[30] Jean-Pierre Tignol; Adrian R. Wadsworth Value functions on simple algebras, and associated graded rings, Springer Monographs in Mathematics, Springer, 2015, xvi+643 pages | DOI | MR | Zbl

[31] Gordon E. Wall The structure of a unitary factor group, Publ. Math., Inst. Hautes Étud. Sci. (1959) no. 1, pp. 7-23 | DOI | Numdam | MR | Zbl

[32] Shianghaw Wang On the commutator group of a simple algebra, Am. J. Math., Volume 72 (1950), pp. 323-334 | DOI | MR | Zbl

[33] Vyacheslav I. Yanchevskiĭ Simple algebras with involutions, and unitary groups, Mat. Sb., N. Ser., Volume 93 (1974) no. 135, pp. 368-380 | MR

[34] Vyacheslav I. Yanchevskiĭ Reduced unitary $K$ -theory and division algebras over Henselian discretely valued fields, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 42 (1978) no. 4, pp. 879-918 | MR | Zbl

[35] Vyacheslav I. Yanchevskiĭ Whitehead groups and groups of $R$ -equivalence classes of linear algebraic groups of non-commutative classical type over some virtual fields, Algebraic groups and arithmetic, Tata Institute of Fundamental Research, 2004, pp. 491-505 | MR | Zbl

Cité par Sources :

Commentaires - Politique