Proceedings of the Japan Academy, Series A, Mathematical Sciences

On some Hasse principles for algebraic groups over global fields

Ngô Thi Ngoan and Nguyêñ Quôć Thǎńg

Full-text: Open access

Abstract

We consider certain local-global principles related with some splitting problems for connected linear algebraic groups over global fields. The main tools are certain reciprocity results due to Prasad and Rapinchuk, Harder’s Hasse principle for homogeneous projective spaces of reductive groups for number fields and their extensions to global function fields.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 5 (2014), 73-78.

Dates
First available in Project Euclid: 1 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.pja/1398949124

Digital Object Identifier
doi:10.3792/pjaa.90.73

Mathematical Reviews number (MathSciNet)
MR3201838

Zentralblatt MATH identifier
1300.11032

Subjects
Primary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10] 14F20: Étale and other Grothendieck topologies and (co)homologies 14L15: Group schemes
Secondary: 14G20: Local ground fields 20G10: Cohomology theory

Keywords
Splitting field tori unipotent groups

Citation

Ngoan, Ngô Thi; Thǎńg, Nguyêñ Quôć. On some Hasse principles for algebraic groups over global fields. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 5, 73--78. doi:10.3792/pjaa.90.73. https://projecteuclid.org/euclid.pja/1398949124


Export citation

References

  • A. Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, 126, Springer, New York, 1991.
  • M. V. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), no. 1, 217–239.
  • B. Conrad, The structure of solvable groups over general fields. (Preprint).
  • B. Conrad, O. Gabber and G. Prasad, Pseudo-reductive groups, New Mathematical Monographs, 17, Cambridge Univ. Press, Cambridge, 2010.
  • J.-L. Colliot-Thélène, P. Gille and R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric fields, Duke Math. J. 121 (2004), 285–341.
  • G. Harder, Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen, Jber. Deutsch. Math.-Verein. 70 (1967/1968), Heft 4, Abt. 1, 182–216.
  • G. Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, J. Reine Angew. Math. 274/275 (1975), 125–138.
  • M. Kneser, Strong approximation, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 187–196, Amer. Math. Soc., Providence, RI, 1966.
  • R. S. Pierce, Associative algebras, Graduate Texts in Mathematics, 88, Springer, New York, 1982.
  • G. Prasad and A. S. Rapinchuk, On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior, Adv. Math. 207 (2006), no. 2, 646–660.
  • J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80.
  • I. Satake, Classification theory of semi-simple algebraic groups, Dekker, New York, 1971.
  • T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, 9, Birkhäuser Boston, Boston, MA, 1998.
  • W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften, 270, Springer, Berlin, 1985.
  • N. Q. Thǎńg, Weak approximation, Brauer and $R$-equivalence in algebraic groups over arithmetical fields, J. Math. Kyoto Univ. 40 (2000), no. 2, 247–291.
  • N. Q. Thǎńg, On Galois cohomology of semisimple groups over local and global fields of positive characteristic, II, Math. Z. 270 (2012), no. 3–4, 1057–1065.
  • J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 33–62, Amer. Math. Soc., Providence, RI, 1966.