Essential dimension, spinor groups and quadratic forms

By P. Brosnan, Zinovy Reichstein, Angelo Vistoli. 2009

58 citations recorded by the source corpus.

We prove that the essential dimension of the spinor group Spin_n grows exponentially with n; in particular, we give a precise formula for this essential dimension when n is not divisible by 4. We use this result to show that the number of 3-fold Pfister forms needed to represent the Witt class of a general quadratic form of rank n with trivial discriminant and Hasse-Witt invariant grows exponentially with n. This paper overlaps with our earlier preprint arXiv:math/0701903 . That preprint has splintered into several parts, which have since acquired a life of their own. In particular, see "Essential dimension of moduli of curves and other algebraic stacks", by the same authors, and "Some consequences of the Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f8d552538fcdc6d0d6fbe0b8934de062fb9e3997

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In Discrete Mathematics and Combinatorics

1. Skip Garibaldi
2. N. Karpenko

In Geometry and Topology

1. Asher Auel
2. Skip Garibaldi
3. N. Karpenko
4. Alexander S. Merkurjev

In Mathematical Physics

1. Skip Garibaldi
2. Asher Auel
3. N. Karpenko
4. Alexander S. Merkurjev

Citations in the corpus

• Open problems on central simple algebras (2010, background): “unds for PGL nstands in interesting constrast with the situtation for Spin : ed p Spinn q is both O ?2 n and Ω 2 n , i.e., is asymptotically bounded both above and below by constants times ? 2 n , by [BRV10]. Essential dimension and crossed products. Let G be a finite group of order n ¥ 2. Let CPAlgn”
• Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties (2009, background): “A G ] 1 8 A pr In the case where G is the group PGL A of automorphisms of a given central simple F - algebra A, for any variety in the class XG (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p.”