Essential dimension, spinor groups, and quadratic forms. (English) Zbl 1252.11034

The paper under review aims to compute the essential dimension of the algebraic groups \(\mathrm{Spin}_n\) over any field of characteristic \(\neq 2\). Application is given to the theory of quadratic forms.
The essential dimension \(\mathrm{ed}(G)\) of an algebraic group \(G\) over a field \(k\) is a measure of complexity of its torsors over field extensions of \(k\). It is defined as the least integer \(n\) such that every \(G\)-torsor over a field extension is defined over some intermediate field of transcendence degree \(\leq n\) over \(k\). See Z. Reichstein [Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures. Hackensack: World Scientific; New Delhi: Hindustan Book Agency. 162–188 (2011; Zbl 1232.14030)] for a survey.
In the paper lower bounds on the essential dimension of the spinor groups \(\mathrm{Spin}_n\) are established which grow exponentially with \(n\). It is shown that in characteristic \(0\) these bounds yield the true value of \(\mathrm{ed}(\mathrm{Spin}_n)\) in case \(n\) is not divisible by \(4\) and \(\geq 15\). The lower bound in case of \(n\equiv 0 \pmod 4\) has recently been improved by V. Chernousov and A.Merkurjev [Essential dimension of spinor and Clifford groups, preprint http://www.math.ucla.edu/~merkurev/papers/i3.pdf].
Torsors for \(\mathrm{Spin}_n\) are closely related to \(n\)-dimensional quadratic forms with trivial discriminant and Hasse-Witt invariant. Consequently the authors use their lower bounds on \(\mathrm{ed}(\mathrm{Spin}_n)\) to show that such quadratic forms are more complex, in high dimensions, than previously expected.
Every quadratic form with trivial discriminant and Hasse-invariant is Witt-equivalent to an orthogonal sum of \(3\)-fold Pfister forms. The following result is established: For any field \(k\) of characteristic \(\neq 2\) there exists a field extension \(K/k\) and an \(n\)-dimensional quadratic form \(q\) over \(K\) with trivial discriminant and Hasse invariant, which is not Witt equivalent to the sum of fewer than \((2^{\frac{n+4}{4}} - n - 2)/7\) of \(3\)-fold Pfister forms over \(K\).
This result stands in contrast with sums of \(1\)-fold and \(2\)-fold Pfister forms where \(n\) such forms always suffice to write down a quadratic form (resp. a quadratic form of trivial discriminant) of even dimension \(n\) up to Witt equivalence.


11E04 Quadratic forms over general fields
11E72 Galois cohomology of linear algebraic groups
15A66 Clifford algebras, spinors


Zbl 1232.14030
Full Text: DOI arXiv Link


[1] J. F. Adams, Lectures on Exceptional Lie Groups, Chicago, IL: University of Chicago Press, 1996. · Zbl 0866.22008
[2] E. M. Andreev and V. L. Popov, ”The stationary subgroups of points in general position in a representation space of a semisimple Lie group,” Funkcional. Anal. i Prilo\vzen., vol. 5, iss. 4, pp. 1-8, 1971. · Zbl 0246.22017 · doi:10.1007/BF01086737
[3] G. Berhuy and G. Favi, ”Essential dimension: a functorial point of view (after A. Merkurjev),” Doc. Math., vol. 8, pp. 279-330, 2003. · Zbl 1101.14324
[4] J. Buhler and Z. Reichstein, ”On the essential dimension of a finite group,” Compositio Math., vol. 106, iss. 2, pp. 159-179, 1997. · Zbl 0905.12003 · doi:10.1023/A:1000144403695
[5] P. Brosnan, ”The essential dimension of a \(g\)-dimensional complex abelian variety is \(2g\),” Transform. Groups, vol. 12, iss. 3, pp. 437-441, 2007. · Zbl 1127.14043 · doi:10.1007/s00031-006-0045-0
[6] P. Brosnan, Z. Reichstein, and A. Vistoli, Essential dimension and algebraic stacks, 2007. · Zbl 1234.14003 · doi:10.4171/JEMS/276
[7] V. Chernousov and J. Serre, ”Lower bounds for essential dimensions via orthogonal representations,” J. Algebra, vol. 305, iss. 2, pp. 1055-1070, 2006. · Zbl 1181.20042 · doi:10.1016/j.jalgebra.2005.10.032
[8] S. Garibaldi, ”Cohomological invariants: exceptional groups and spin groups,” Mem. Amer. Math. Soc., vol. 200, 2009. · Zbl 1191.11009
[9] S. Garibaldi, A. Merkurjev, and J. Serre, Cohomological Invariants in Galois Cohomology, Providence, RI: Amer. Math. Soc., 2003. · Zbl 1159.12311
[10] N. A. Karpenko and A. S. Merkurjev, ”Essential dimension of finite \(p\)-groups,” Invent. Math., vol. 172, iss. 3, pp. 491-508, 2008. · Zbl 1200.12002 · doi:10.1007/s00222-007-0106-6
[11] M. Knus, A. Merkurjev, M. Rost, and J. Tignol, The Book of Involutions, Providence, RI: Amer. Math. Soc., 1998. · Zbl 0955.16001
[12] T. Y. Lam, The Algebraic Theory of Quadratic Forms, W. A. Benjamin, Reading, MA, 1973. · Zbl 0259.10019
[13] A. Merkurjev, ”Essential dimension,” in Quadratic Forms - Algebra, Arithmetic , and Geometry, , 2009, vol. 493, pp. 299-326. · Zbl 1188.14006
[14] A. Meyer and Z. Reichstein, Some consequences of the Karpenko-Merkurjev theorem. · Zbl 1277.20059
[15] R. Parimala, V. Suresh, and J. -P. Tignol, ”On the Pfister number of quadratic forms,” in Quadratic Forms - Algebra, Arithmetic, and Geometry, Baeza, R., Chan, W. K., Hoffman, D. W., and Schulze-Pillot, R., Eds., , 2009, vol. 493, pp. 327-338. · Zbl 1208.11055
[16] A. M. Popov, ”Finite stationary subgroups in general position of simple linear Lie groups,” Trudy Moskov. Mat. Obshch., vol. 48, pp. 7-59, 263, 1985. · Zbl 0661.22009
[17] V. L. Popov and E. B. Vinberg, Algebraic Geometry. IV, New York: Springer-Verlag, 1994.
[18] Z. Reichstein, ”On the notion of essential dimension for algebraic groups,” Transform. Groups, vol. 5, iss. 3, pp. 265-304, 2000. · Zbl 0981.20033 · doi:10.1007/BF01679716
[19] R. W. Richardson, ”Conjugacy classes of \(n\)-tuples in Lie algebras and algebraic groups,” Duke Math. J., vol. 57, iss. 1, pp. 1-35, 1988. · Zbl 0685.20035 · doi:10.1215/S0012-7094-88-05701-8
[20] M. Rost, ”A descent property for Pfister forms,” J. Ramanujan Math. Soc., vol. 14, iss. 1, pp. 55-63, 1999. · Zbl 1059.11033
[21] Z. Reichstein and B. Youssin, ”Essential dimensions of algebraic groups and a resolution theorem for \(G\)-varieties,” Canad. J. Math., vol. 52, iss. 5, pp. 1018-1056, 2000. · Zbl 1044.14023 · doi:10.4153/CJM-2000-043-5
[22] J. A. Wood, ”Spinor groups and algebraic coding theory,” J. Combin. Theory Ser. A, vol. 51, iss. 2, pp. 277-313, 1989. · Zbl 0704.22010 · doi:10.1016/0097-3165(89)90053-8
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.