Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
[Principe local-global pour les formes quadratiques sur les corps de fractions d’anneaux henséliens de dimension deux]
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2131-2143.

Soit R un anneau local intègre de dimension 2, normal, excellent et hensélien dans lequel 2 est inversible. Soient L son corps de fractions et k son corps résiduel. Soit Ω R l’ensemble des valuations discrètes de rang 1 de L correspondant aux points de codimension 1 des modèles propres réguliers de SpecR. On démontre qu’une forme quadratique q sur L satisfait le principe local-global par rapport à Ω R dans les deux cas suivants : (1) q est de rang 3 ou 4 ; (2) q est de rang 5 et R=A[[y]], où A est un anneau de valuation discrète complet, avec une condition sur le corps résiduel k qui est satisfaite lorsque k est C 1 .

Let R be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let L and k be its fraction field and residue field respectively. Let Ω R be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of SpecR. We prove that a quadratic form q over L satisfies the local-global principle with respect to Ω R in the following two cases: (1) q has rank 3 or 4; (2) q has rank 5 and R=A[[y]], where A is a complete discrete valuation ring with a not too restrictive condition on the residue field k, which is satisfied when k is C 1 .

DOI : 10.5802/aif.2745
Classification : 11E04, 11E08, 11D88, 14G99
Keywords: 2-dimensional local ring, local-global principle, quadratic forms, complete local domain
Mot clés : anneau local de dimension 2, principe local-global, formes quadratiques, anneau local complet
HU, Yong 1

1 Université Paris-Sud, Département de Mathématiques, Bâtiment 425, 91405, Orsay Cedex (France)
@article{AIF_2012__62_6_2131_0,
     author = {HU, Yong},
     title = {Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains},
     journal = {Annales de l'Institut Fourier},
     pages = {2131--2143},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     doi = {10.5802/aif.2745},
     mrnumber = {3060754},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2745/}
}
TY  - JOUR
AU  - HU, Yong
TI  - Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 2131
EP  - 2143
VL  - 62
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2745/
DO  - 10.5802/aif.2745
LA  - en
ID  - AIF_2012__62_6_2131_0
ER  - 
%0 Journal Article
%A HU, Yong
%T Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
%J Annales de l'Institut Fourier
%D 2012
%P 2131-2143
%V 62
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2745/
%R 10.5802/aif.2745
%G en
%F AIF_2012__62_6_2131_0
HU, Yong. Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2131-2143. doi : 10.5802/aif.2745. http://www.numdam.org/articles/10.5802/aif.2745/

[1] Choi, M. D.; Dai, Z. D.; Lam, T. Y.; Reznick, B. The Pythagoras number of some affine algebras and local algebras, J. Reine Angew. Math., Volume 336 (1982), pp. 45-82 | DOI | MR | Zbl

[2] Colliot-Thélène, J.-L.; Ojanguren, M.; Parimala, R. Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) (Tata Inst. Fund. Res. Stud. Math.), Volume 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 185-217 | MR | Zbl

[3] Colliot-Thélène, M.; Parimala, R.; Suresh, V. Patching and local-global principle for homogeneous spaces over function fields of p-adic curves., Comment. Math. Helv. (to appear)

[4] Harbater, David; Hartmann, Julia; Krashen, Daniel Applications of patching to quadratic forms and central simple algebras, Invent. Math., Volume 178 (2009) no. 2, pp. 231-263 | DOI | MR

[5] Heath-Brown, D. R. Zeros of systems of 𝔭-adic quadratic forms, Compos. Math., Volume 146 (2010) no. 2, pp. 271-287 | DOI | MR | Zbl

[6] Jaworski, Piotr On the strong Hasse principle for fields of quotients of power series rings in two variables, Math. Z., Volume 236 (2001) no. 3, pp. 531-566 | DOI | MR | Zbl

[7] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998 (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | DOI | MR | Zbl

[8] Lam, T. Y. Introduction to quadratic forms over fields, Graduate Studies in Mathematics, 67, American Mathematical Society, Providence, RI, 2005 | MR | Zbl

[9] Leep, D. The u-invariant of p-adic function fields (Preprint)

[10] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6, Oxford University Press, Oxford, 2002 (Translated from the French by Reinie Erné, Oxford Science Publications) | MR | Zbl

[11] Parimala, Raman; Suresh, V. The u-invariant of the function fields of p-adic curves, Ann. of Math. (2), Volume 172 (2010) no. 2, pp. 1391-1405 | DOI | MR | Zbl

[12] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York, 1979 (Translated from the French by Marvin Jay Greenberg) | MR | Zbl

[13] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997 | MR | Zbl

Cité par Sources :