Comptes Rendus
Article de recherche - Géométrie algébrique, Théorie des nombres
A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation
Yisheng Tian1
1 Institute for Advanced Study in Mathematics, Harbin Institute of Technology, 150001 Harbin, China
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 257-264.

We show that certain Tate–Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.547
Yisheng Tian 1

1 Institute for Advanced Study in Mathematics, Harbin Institute of Technology, 150001 Harbin, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2024__362_G3_257_0,
     author = {Yisheng Tian},
     title = {A {Comparison} of {Cohomological} {Obstructions} to the {Hasse} {Principle} and to {Weak} {Approximation}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {257--264},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.547},
     language = {en},
}
TY  - JOUR
AU  - Yisheng Tian
TI  - A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 257
EP  - 264
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.547
LA  - en
ID  - CRMATH_2024__362_G3_257_0
ER  - 
%0 Journal Article
%A Yisheng Tian
%T A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation
%J Comptes Rendus. Mathématique
%D 2024
%P 257-264
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.547
%G en
%F CRMATH_2024__362_G3_257_0
Yisheng Tian. A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 257-264. doi : 10.5802/crmath.547. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.547/

[1] Y. Cao Troisième groupe de cohomologie non ramifiée des torseurs universels sur les surfaces rationnelles, Épijournal de Géom. Algébr., EPIGA, Volume 2 (2018), 12 | Zbl

[2] J.-L. Colliot-Thélène Birational invariants, purity and the Gersten conjecture, K-theory and algebraic geometry: connections with quadratic forms and division algebras, Santa Barbara, CA (Proceedings of Symposia in Pure Mathematics), Volume 58, American Mathematical Society (1995), pp. 1-64 | MR | Zbl

[3] J.-L. Colliot-Thélène Descente galoisienne sur le second groupe de Chow: mise au point et applications, Doc. Math. (2015), pp. 195-220 | MR | Zbl

[4] J.-L. Colliot-Thélène; D. Harari; A. Skorobogatov Compactification équivariante d’un tore (d’après Brylinski et Künnemann), Expo. Math., Volume 23 (2005) no. 2, pp. 161-170 | DOI | Zbl

[5] L. Fu Etale cohomology theory, Nankai Tracts in Mathematics, 13, World Scientific, 2011 | DOI | Zbl

[6] W. Fulton Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 2, Springer, 1984 | DOI | Zbl

[7] D. Harari; C. Scheiderer; T. Szamuely Weak approximation for tori over p-adic function fields, Int. Math. Res. Not. (2015) no. 10, pp. 2751-2783 | DOI | MR | Zbl

[8] D. Harari; T. Szamuely Local-global questions for tori over p-adic function fields, J. Algebr. Geom., Volume 25 (2016) no. 3, pp. 571-605 | DOI | MR | Zbl

[9] M. Kashiwara; P. Schapira Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, 332, Springer, 2006 | DOI | Zbl

[10] J. Kollár Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, 2007 | Zbl

[11] Y. Tian Obstructions to weak approximation for reductive groups over p-adic function fields, J. Number Theory, Volume 220 (2021), pp. 128-162 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Principe de Hasse pour les intersections de deux quadriques

Olivier Wittenberg

C. R. Math (2006)

On the genus of a division algebra

Vladimir I. Chernousov; Andrei S. Rapinchuk; Igor A. Rapinchuk

C. R. Math (2012)

Vanishing of H1 for Dedekind rings and applications to loop algebras

Arturo Pianzola

C. R. Math (2005)