×

Isotropy of certain quadratic forms of dimensions 7 and 8 over the function field of a quadric. (Isotropie de certaines formes quadratiques de dimensions 7 et 8 sur le corps des fonctions d’une quadrique.) (French) Zbl 0865.11030

This is the first of a series of three papers announced by the author in C. R. Acad. Sci., Paris, Sér. I 323, No. 5, 495-499 (1996; see Zbl 0865.11030 above).
Let \(\varphi\) be an \(F\)-anisotropic quadratic form and \(\dim\varphi =8\), \(\text{discr} \varphi =1\). Assume the index of the Clifford algebra \(C(\varphi)\) is \(\leq 4\). A complete characterization is given for quadratic forms \(\psi\) with the property that \(\varphi\) becomes isotropic over the function field \(F(\psi)\) of the projective quadric \(\psi=0\) (except in the case when \(\dim \psi=4\) and \(\psi\) is not similar to a Pfister form). There is a corollary for the case where \(\varphi\) is replaced with a form of dimension 7. The proofs use not only Merkurjev’s and Merkurjev and Suslin’s theorems that \(e^2\) and \(e^3\) are isomorphisms, but also the determination of the cohomological kernels \(\ker(H^2F \to H^2K)\) and \(\ker (H^3F \to H^3K)\), where \(K\) is the function field of the Severi-Brauer variety of \(C(\varphi)\).

MSC:

11E04 Quadratic forms over general fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. A. Amitsur, Generic splitting fields of central simple algebras , Ann. of Math. (2) 62 (1955), 8-43. JSTOR: · Zbl 0066.28604 · doi:10.2307/2007098
[2] J. Kr. Arason, Cohomologische invarianten quadratischer Formen , J. Algebra 36 (1975), no. 3, 448-491. · Zbl 0314.12104 · doi:10.1016/0021-8693(75)90145-3
[3] M. Artin, Brauer-Severi varieties , Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin, 1982, pp. 194-210. · Zbl 0536.14006 · doi:10.1007/BFb0092235
[4] R. Elman and T. Y. Lam, Pfister forms and \(K\)-theory of fields , J. Algebra 23 (1972), 181-213. · Zbl 0246.15029 · doi:10.1016/0021-8693(72)90054-3
[5] R. Elman, T. Y. Lam, and A. R. Wadsworth, Amenable fields and Pfister extensions , Conference on Quadratic Forms-1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976), Queen’s Univ., Kingston, Ont., 1977, pp. 445-492. · Zbl 0385.12010
[6] D. W. Hoffmann, Isotropy of \(5\)-dimensional quadratic forms over the function field of a quadric , \(K\)-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 217-225. · Zbl 0824.11023
[7] D. W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric , Math. Z. 220 (1995), no. 3, 461-476. · Zbl 0840.11017 · doi:10.1007/BF02572626
[8] D. W. Hoffmann, On quadratic forms of height two and a theorem of Wadsworth , prépublication, 1994.
[9] D. W. Hoffmann, On \(6\)-dimensional quadratic forms isotropic over the function field of a quadric , Comm. Algebra 22 (1994), no. 6, 1999-2014. · Zbl 0804.11031 · doi:10.1080/00927879408824952
[10] B. Kahn, A descent problem for quadratic forms , Duke Math. J. 80 (1995), no. 1, 139-155. · Zbl 0858.11024 · doi:10.1215/S0012-7094-95-08006-5
[11] B. Kahn, Formes quadratiques de hauteur et de degré \(2\) , à paraître dans Indag. Math. (N.S.).
[12] M. Knebusch, Generic splitting of quadratic forms. I , Proc. London Math. Soc. (3) 33 (1976), no. 1, 65-93. · Zbl 0351.15016 · doi:10.1112/plms/s3-33.1.65
[13] M. Knebusch, Generic splitting of quadratic forms. II , Proc. London Math. Soc. (3) 34 (1977), no. 1, 1-31. · Zbl 0359.15013 · doi:10.1112/plms/s3-34.1.1
[14] T. Y. Lam, The algebraic theory of quadratic forms , 2éme éd. ed., Mathematics Lecture Note Series, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1980. · Zbl 0437.10006
[15] D. Leep, Function fields results , 1989, notes manuscrites prises par T. Y. Lam. · Zbl 0698.12013
[16] A. S. Merkurev, Simple algebras and quadratic forms , Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, 218-224, traduction anglaise dans Math. USSR Izv. 38 (1992), 215-221. · Zbl 0733.12008
[17] A. S. Merkurev, On the norm residue symbol of degree \(2\) , Dokl. Akad. Nauk SSSR 261 (1981), no. 3, 542-547, (en russe), traduction anglaise dans Soviet Math. Dokl. 24 (1981), 546-551. · Zbl 0496.16020
[18] A. S. Merkurev and A. A. Suslin, Norm residue homomorphism of degree three [L’homomorphisme de résidu normique de degré\(3\)] , Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 339-356, (en russe), traduction anglaise dans Math. USSR Izv. 36 (1991), 349-367. · Zbl 0711.19003
[19] J. W. Milnor, Algebraic \(K\)-theory and quadratic forms , Invent. Math. 9 (1969/1970), 318-344. · Zbl 0199.55501 · doi:10.1007/BF01425486
[20] E. Peyre, Products of Severi-Brauer varieties and Galois cohomology , \(K\)-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 369-401. · Zbl 0837.14006
[21] M. Rost, Hilbert’s theorem 90 for \(K^M_3\) for degree-two extensions , prépublication, Regensburg, 1986.
[22] M. Rost, On quadratic forms isotropic over the function field of a conic , Math. Ann. 288 (1990), no. 3, 511-513. · Zbl 0698.12009 · doi:10.1007/BF01444545
[23] W. Scharlau, Quadratic and Hermitian forms , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. · Zbl 0584.10010
[24] A. Schofield and M. Van den Bergh, The index of a Brauer class on a Brauer-Severi variety , Trans. Amer. Math. Soc. 333 (1992), no. 2, 729-739. · Zbl 0778.12004 · doi:10.2307/2154058
[25] J.-P. Tignol, Réduction de l’indice d’une algèbre simple centrale sur le corps des fonctions d’une quadrique , Bull. Soc. Math. Belg. Sér. A 42 (1990), no. 3, 735-745. · Zbl 0736.12002
[26] A. R. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields , Trans. Amer. Math. Soc. 208 (1975), 352-358. JSTOR: · Zbl 0336.15013 · doi:10.2307/1997291
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.