On quadratic forms of height two and a theorem of Wadsworth. (English) Zbl 0867.11024
For quadratic forms \(\varphi\) and \(\psi\) over a field \(F\) of characteristic different from two, the function fields \(F(\varphi)\) and \(F(\psi)\) are equivalent (i.e. there are \(F\)-places from each function field to the other) if and only if \(\varphi\) is isotropic over \(F(\psi)\) and \(\psi\) is isotropic over \(F(\varphi)\). This was proved by M. Knebusch in 1976. Equivalence of function fields of quadratic forms is, in general, weaker than their isomorphism, and the cases where the two coincide are of special interest. A third property involved is the similarity of the quadratic forms in question. Here a pioneering work was done by A. R. Wadsworth [Trans. Am. Math. Soc. 208, 352-358 (1975; Zbl 0336.15013)], who proved that for forms of dimension 4 and discriminant 1 the isomorphism of function fields of the forms implies similarity of the forms.
Combining the results of several authors on the interdependence of the equivalence and isomorphism of function fields of quadratic forms and of similarity of quadratic forms (A. R. Wadsworth, D. W. Leep, and the author himself) the author obtains a list of forms \(\varphi\) for which the three properties are equivalent for \(\varphi\) and any quadratic form \(\psi\) with \(\dim\psi=\dim\varphi\). This list consists of Pfister neighbors of codimension 0 or 1, and of forms of dimension at most 6 satisfying some additional conditions. This is the starting point for the author’s first main result.
It is shown that if an anisotropic form \(\varphi\) of dimension \(2^n\), \(n\geq 2\), can be written as the product of an \((n-2)\)-fold Pfister form and a four-dimensional form, then for any quadratic form \(\psi\) of dimension \(2^n\) the following are equivalent: (1) \(\varphi\) is similar to \(\psi\). (2) The function fields \(F(\varphi)\) and \(F(\psi)\) are isomorphic. (3) The function fields \(F(\varphi)\) and \(F(\psi)\) are equivalent.
The second main result asserts that (2) and (3) are equivalent when \(\varphi\) is an anisotropic form of height 2 and degree 1 or 2, \(\psi\) is an anisotropic form, and \(\dim\varphi=\dim\psi\).
Combining the results of several authors on the interdependence of the equivalence and isomorphism of function fields of quadratic forms and of similarity of quadratic forms (A. R. Wadsworth, D. W. Leep, and the author himself) the author obtains a list of forms \(\varphi\) for which the three properties are equivalent for \(\varphi\) and any quadratic form \(\psi\) with \(\dim\psi=\dim\varphi\). This list consists of Pfister neighbors of codimension 0 or 1, and of forms of dimension at most 6 satisfying some additional conditions. This is the starting point for the author’s first main result.
It is shown that if an anisotropic form \(\varphi\) of dimension \(2^n\), \(n\geq 2\), can be written as the product of an \((n-2)\)-fold Pfister form and a four-dimensional form, then for any quadratic form \(\psi\) of dimension \(2^n\) the following are equivalent: (1) \(\varphi\) is similar to \(\psi\). (2) The function fields \(F(\varphi)\) and \(F(\psi)\) are isomorphic. (3) The function fields \(F(\varphi)\) and \(F(\psi)\) are equivalent.
The second main result asserts that (2) and (3) are equivalent when \(\varphi\) is an anisotropic form of height 2 and degree 1 or 2, \(\psi\) is an anisotropic form, and \(\dim\varphi=\dim\psi\).
Reviewer: K.Szymiczek (Tychy)
MSC:
| 11E04 | Quadratic forms over general fields |
| 12F20 | Transcendental field extensions |
| 11E81 | Algebraic theory of quadratic forms; Witt groups and rings |
Keywords:
quadratic forms of height two; equivalence of function fields of quadratic forms; isomorphism of function fields; similarity of quadratic forms; Pfister neighbors; anisotropic form; Pfister formCitations:
Zbl 0336.15013References:
| [1] | Jón Kr. Arason and Manfred Knebusch, Über die Grade quadratischer Formen, Math. Ann. 234 (1978), no. 2, 167 – 192 (German). · Zbl 0362.10021 · doi:10.1007/BF01420967 |
| [2] | H. Ahmad and J. Ohm, Function fields of Pfister neighbors, J. Algebra. 178 (1995), 653–664. CMP 1996:4 |
| [3] | Richard Elman and T. Y. Lam, Pfister forms and \?-theory of fields, J. Algebra 23 (1972), 181 – 213. · Zbl 0246.15029 · doi:10.1016/0021-8693(72)90054-3 |
| [4] | Richard Elman and T. Y. Lam, Quadratic forms and the \?-invariant. II, Invent. Math. 21 (1973), 125 – 137. · Zbl 0267.10029 · doi:10.1007/BF01389692 |
| [5] | Richard Elman, T. Y. Lam, and Adrian 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. With an appendix ”Excellence of \?(\?)/\? for 2-fold Pfister forms” by J. K. Arason. Queen’s Papers in Pure and Appl. Math., No. 46. · Zbl 0385.12010 |
| [6] | R. Elman, T. Y. Lam, and A. R. Wadsworth, Function fields of Pfister forms, Invent. Math. 51 (1979), no. 1, 61 – 75. · Zbl 0395.10028 · doi:10.1007/BF01389912 |
| [7] | Robert W. Fitzgerald, Quadratic forms of height two, Trans. Amer. Math. Soc. 283 (1984), no. 1, 339 – 351. · Zbl 0512.10011 |
| [8] | D. W. Hoffmann, Function Fields of Quadratic Forms, Ph.D. thesis, University of California, Berkeley, California 1992. |
| [9] | ——, Isotropy of \(5\)-dimensional quadratic forms over the function field of a quadric, Proceedings of the 1992 Santa Barbara Summer Research Institute on Quadratic Forms and Division Algebras, Proc. Symp. Pure Math. 58.2 (1995), 217–225. CMP 1995:11 |
| [10] | Detlev 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 |
| [11] | ——, Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995), 461–476. CMP 1996:4 |
| [12] | D. W. Hoffmann, D. W. Lewis, and J. Van Geel, Minimal forms for function fields of conics, Proceedings of the 1992 Santa Barbara Summer Research Institute on Quadratic Forms and Division Algebras, Proc. Symp. Pure Math. 58.2 (1995), 227–237. CMP 1995:11 · Zbl 0824.11024 |
| [13] | J. Hurrelbrink and U. Rehmann, Splitting patterns of quadratic forms, Math. Nachr. 176 (1995), 111–127. CMP 1996:4 · Zbl 0876.11017 |
| [14] | O.T. Izhboldin, On the nonexcellence of field extensions \(F(\pi )/F\), preprint (1995). · Zbl 0857.11017 |
| [15] | B. Kahn, Formes quadratiques de hauteur et de degré \(2\), to appear in: Indag. Math. · Zbl 0866.11030 |
| [16] | Manfred 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 |
| [17] | Manfred 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 |
| [18] | T. Y. Lam, The algebraic theory of quadratic forms, W. A. Benjamin, Inc., Reading, Mass., 1973. Mathematics Lecture Note Series. T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. Revised second printing; Mathematics Lecture Note Series. · Zbl 0259.10019 |
| [19] | D. Leep, Function field results, handwritten notes taken by T. Y. Lam (1989). |
| [20] | Pasquale Mammone and Daniel B. Shapiro, The Albert quadratic form for an algebra of degree four, Proc. Amer. Math. Soc. 105 (1989), no. 3, 525 – 530. · Zbl 0669.10042 |
| [21] | J. Ohm, The Zariski problem for function fields of quadratic forms, to appear in: Proc. Amer. Math. Soc. · Zbl 0859.11027 |
| [22] | Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. · Zbl 0584.10010 |
| [23] | Adrian R. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields, Trans. Amer. Math. Soc. 208 (1975), 352 – 358. · Zbl 0336.15013 |
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.
