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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.

MSC:

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

Citations:

Zbl 1232.14030
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References:

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