Proceedings of the Japan Academy, Series A, Mathematical Sciences

On some Hasse principles for algebraic groups over global fields. II

Ngô Thị Ngoan and Nguyêñ Quôć Thǎńg

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Abstract

In this paper, we prove the validity of the cohomological Hasse principle for $\mathrm{H}^{1}$ of semisimple simply connected algebraic groups defined over infinite algebraic extensions of global fields and also some local–global principles for (skew-)hermitian forms defined over such fields.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 8 (2014), 107-112.

Dates
First available in Project Euclid: 3 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.pja/1412341993

Digital Object Identifier
doi:10.3792/pjaa.90.107

Mathematical Reviews number (MathSciNet)
MR3266743

Zentralblatt MATH identifier
1300.11032

Subjects
Primary: 11E08: Quadratic forms over local rings and fields 11E12: Quadratic forms over global rings and fields 11E39: Bilinear and Hermitian forms 11E72: Galois cohomology of linear algebraic groups [See also 20G10]
Secondary: 20G25: Linear algebraic groups over local fields and their integers 20G30: Linear algebraic groups over global fields and their integers

Keywords
Hasse principle algebraic groups

Citation

Ngoan, Ngô Thị; Thǎńg, Nguyêñ Quôć. On some Hasse principles for algebraic groups over global fields. II. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 107--112. doi:10.3792/pjaa.90.107. https://projecteuclid.org/euclid.pja/1412341993


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References

  • V. Andriychuk, Some applications of Hasse principle for pseudoglobal fields, Algebra Discrete Math. 2004, no. 2, 1–8.
  • E. Artin and J. Tate, Class field theory, 2nd ed., Advanced Book Classics, Addison-Wesley, Redwood City, CA, 1990.
  • F. Bruhat and J. Tits, Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 671–698.
  • I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, 2nd ed., Translations of Mathematical Monographs, 121, Amer. Math. Soc., Providence, RI, 2002.
  • G. Harder, Über die Galoiskohomologie halbeinfacher Matrizengruppen, I. Math. Z. Bd. 90 (1965), 404–428.
  • G. Harder, Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, J. Reine Angew. Math. 274/275 (1975), 125–138.
  • Y. Kawada, Class formulations. V. Infinite extension of the $p$-adic field or the rational field, J. Math. Soc. Japan 12 (1960), 34–64.
  • M. Kneser, Lectures on Galois cohomology of classical groups, Tata Inst. Fund. Res., Bombay, 1969.
  • K. Kozioł and M. Kula, Witt rings of infinite algebraic extensions of global fields, Ann. Math. Sil. 12 (1998), 131–139.
  • D. König, Sur les correspondances multivoques des ensembles, Fund. Math. 8 (1926), 114–134.
  • J. S. Milne, Arithmetic duality theorems, 2nd ed., BookSurge, LLC, Charleston, SC, 2006.
  • M. Moriya, Theorie der algebraischen Zahlkörper unendlichen Grades, Journ. Fac. Sci., Hokkaido Imp. Univ. 3 (1935), 107–190.
  • M. Moriya, Divisionsalgebren über einem $\mathfrak{p}$-adischen Zahlkörper eines unendlichen algebraischen Zahlkörpers, Proc. Imp. Acad. 12 (1936), no. 7, 183–184.
  • N. T. Ngoan and N. Q. Thǎńg, On some Hasse principles for algebraic groups over global fields, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 5, 73–78.
  • N. T. Ngoan and N. Q. Thǎńg, On the arithmetic of algebraic groups over infinite global fields. (in preparation).
  • V. Platonov and A. Rapinchuk, Algebraic groups and number theory, translated from the 1991 Russian original by Rachel Rowen, Pure and Applied Mathematics, 139, Academic Press, Boston, MA, 1994.
  • 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.
  • W. Scharlau, Klassifikation hermitescher Formen über lokalen Körpern, Math. Ann. 186 (1970), 201–208.
  • W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften, 270, Springer, Berlin, 1985.
  • O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, Amer. Math. Soc., New York, NY, 1950.
  • J.-P. Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, 5, Springer, Berlin, 1994. (English translation: Galois cohomology, Springer, 1997).
  • M. A. Tsfasman and S. G. Vlăduţ, Infinite global fields and the generalized Brauer-Siegel theorem, Mosc. Math. J. 2 (2002), no. 2, 329–402.
  • T. Tsukamoto, On the local theory of quaternionic anti-hermitian forms, J. Math. Soc. Japan 13 (1961), 387–400.