Abstract
Essential dimension is an invariant of algebraic groups G over a field F that measures the complexity of G-torsors over field extensions of F. We use theorems of N. Karpenko about the incompressibility of Severi-Brauer varieties and quadratic Weil transfers of Severi-Brauer varieties to compute the essential dimension of some closed subgroups of R K/F (GL 1(A)), where A is a central division K-algebra of prime power degree and K/F is a separable field extension of degree ≤ 2. In particular, we determine the essential dimension of the group Sim(A, σ) of similitudes of (A, σ), where σ is an F-involution on A, and the essential dimension of the normalizer \(N_{GL_1 (A)} \left( {GL_1 \left( B \right)} \right)\), where B is a separable subalgebra of A.
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Lötscher, R. Essential dimension of involutions and subalgebras. Isr. J. Math. 192, 325–346 (2012). https://doi.org/10.1007/s11856-012-0037-9
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DOI: https://doi.org/10.1007/s11856-012-0037-9