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Cohomological invariants of odd degree Jordan algebras

Published online by Cambridge University Press:  01 September 2008

MARK L. MacDONALD
MARK L. MacDONALD*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB. e-mail: [email protected]
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Abstract

In this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n ≥ 3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp2n) = n + 1 for n odd.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 2 , September 2008 , pp. 295 - 303
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Berhuy, G. and Favi, G.. Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math. 8 (2003), 279–330.CrossRefGoogle Scholar
[2]Chernousov, V. and J.–P. Serre. Lower bounds for essential dimensions via orthogonal representations. J. Algebra 305 (2006), 1055–1070.Google Scholar
[3]Delzant, A.. Définition des classes de Stiefel–Whitney d'un module quadratique sur un corps de caractéristique différente de 2. C. R. Acad. Sci. Paris 255 (1962), 1366–1368.Google Scholar
[4]Garibaldi, S.. Cohomological invariants: exceptional groups and spin groups. Mem. Amer. Math. Soc., to appear.Google Scholar
[5]Garibaldi, S., Merkurjev, A. and Serre, J.–P.. Cohomological invariants in Galois cohomology. AMS University Lecture Series 28 (American Mathematical Society, 2003).Google Scholar
[6]Garibaldi, S., Parimala, R. and Tignol, J.-P.. Discriminant of symplectic involutions. Preprint, www.mathematik.uni-bielefeld.de/lag/man/264.html.Google Scholar
[7]Garibaldi, S., Quéguiner-Mathieu, A. and Tignol, J.-P.. Involutions and trace forms on exterior powers of a central simple algebra. Doc. Math. 6 (2001), 99–120.CrossRefGoogle Scholar
[8]Haile, D., Knus, M.-A., Rost, M. and Tignol, J.-P.. Algebras of odd degree with involution, trace forms and dihedral extensions. Israel J. Math. 96 (1996), 299–340.Google Scholar
[9]Jacobson, N.. Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq. Publ. vol. XXXIX (American Mathematical Society, 1968).Google Scholar
[10]Kordonski, V. È.. On the essential dimension and Serre's conjecture II for exceptional groups. (Russian. Russian summary) Mat. Zametki (4) 68 (2000), 539–547; English translation in Math. Notes (3–4) 68 (2000), 464–470.Google Scholar
[11]Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P.. The Book of Involutions. Colloquium Publications (American Mathematical Society, 1998).CrossRefGoogle Scholar
[12]Lemire, N.. Essential dimension of algebraic groups and integral representations of Weyl groups. Transform. Groups 9 (2004), 337–379.Google Scholar
[13]McCrimmon, K.. A Taste of Jordan algebras. Universitext (Springer, 2004).Google Scholar
[14]Milnor, J.. Algebraic K-theory and quadratic forms. Invent. Math. 9 (1970), 318–344.Google Scholar
[15]Reichstein, Z. and Youssin, B.. Essential dimensions of algebraic groups and a resolution theorem for G-varieties (with an appendix by János Kollár and Endre Szabó). Canad. J. Math. 52 (2000), 1018–1056.Google Scholar
[16]Rost, M.. A descent property for Pfister forms. J. Ramanujan Math. Soc. 14 (1999), 55–63.Google Scholar
[17]Scharlau, W.. Quadratische Formen und Galois-Cohomologie. Invent. Math. 4 (1967), 238–264.Google Scholar
[18]Scharlau, W.. Quadratic and hermitian forms. Grundlehren der Mathematischen Wissenschaften, vol. 270 (Springer-Verlag, 1985).Google Scholar
[19]Serre, J.-P.. Cohomologie galoisienne: progrès et problèmes. Astérisque 227 (1995), 229–257.Google Scholar
[20]Springer, T. A. and Veldkamp, F. D.. Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics (Springer, 2000).Google Scholar
[21]Wadsworth, A. R. and Shapiro, D. B.. On multiples of round and Pfister forms. Math. Z. 157 (1977), 53–62.CrossRefGoogle Scholar