Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2131-2143

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 .

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 .

DOI : https://doi.org/10.5802/aif.2745
Classification:  11E04,  11E08,  11D88,  14G99
Keywords: 2-dimensional local ring, local-global principle, quadratic forms, complete local domain
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2131-2143},
     doi = {10.5802/aif.2745},
     mrnumber = {3060754},
     zbl = {pre06159908},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 62 (2012) no. 6, pp. 2131-2143. doi : 10.5802/aif.2745. http://www.numdam.org/item/AIF_2012__62_6_2131_0/

[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., Tome 336 (1982), pp. 45-82 | Article | MR 671321 | Zbl 0499.12018

[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., Bombay (Tata Inst. Fund. Res. Stud. Math.) Tome 16 (2002), pp. 185-217 | MR 1940669 | Zbl 1055.14019

[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., Tome 178 (2009) no. 2, pp. 231-263 | Article | MR 2545681 | Zbl pre05627032

[5] Heath-Brown, D. R. Zeros of systems of 𝔭-adic quadratic forms, Compos. Math., Tome 146 (2010) no. 2, pp. 271-287 | Article | MR 2601629 | Zbl 1194.11047

[6] Jaworski, Piotr On the strong Hasse principle for fields of quotients of power series rings in two variables, Math. Z., Tome 236 (2001) no. 3, pp. 531-566 | Article | MR 1821304 | Zbl 1009.11027

[7] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 134 (1998) (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | Article | MR 1658959 | Zbl 0926.14003

[8] Lam, T. Y. Introduction to quadratic forms over fields, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 67 (2005) | MR 2104929 | Zbl 1068.11023

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

[10] Liu, Qing Algebraic geometry and arithmetic curves, Oxford University Press, Oxford, Oxford Graduate Texts in Mathematics, Tome 6 (2002) (Translated from the French by Reinie Erné, Oxford Science Publications) | MR 1917232 | Zbl 0996.14005

[11] Parimala, Raman; Suresh, V. The u-invariant of the function fields of p-adic curves, Ann. of Math. (2), Tome 172 (2010) no. 2, pp. 1391-1405 | Article | MR 2680494 | Zbl 1208.11053

[12] Serre, Jean-Pierre Local fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 67 (1979) (Translated from the French by Marvin Jay Greenberg) | MR 554237 | Zbl 0423.12016

[13] Washington, Lawrence C. Introduction to cyclotomic fields, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 83 (1997) | MR 1421575 | Zbl 0484.12001