On the essential dimension of cyclic groups
The essential dimension of an algebraic structure is a numerical invariant that measures its complexity. Informally, the essential dimension over a field \(F\) is the smallest number of algebraically independent parameters, needed to define the structure over an extension of \(F\) see \textit{G. Berhuy} and \textit{G. Favi}, [Doc. Math., J. DMV 8, 279--330 (2003; Zbl 1101.14324)]. The essential dimension \(\mathrm{ed}(G)\) of a finite group \(G\) measures the complexity of the category of \(G\)-torsors. When \(G\) is a cyclic \(p\)-group and \(F\) contains a primitive \(p\)-th root of unity, \(\mathrm{ed}(G)\) has been computed by \textit{M. Florence} (see [Invent. Math. 171, No. 1, 175--189 (2008; Zbl 1136.14035)], \textit{N. A. Karpenko} and \textit{A. S. Merkurjev}, [Invent. Math. 172, No. 3, 491--508 (2008; Zbl 1200.12002)]). The main result of the paper under review finds an upper bound for \(\mathrm{ed}(G)\). The paper also obtains a lower bound for \(\mathrm{ed}(G)\) in terms of the canonical dimension \(\mathrm{cdim}(A)\) of some central division \(F\)-algebra \(A\). Both results are obtained under the hypothesis that the field \(F\) contains a primitive \(\pi \)-th root of unity, for each prime divisor \(\pi \) of the order of \(G\). It is also proved that, under a conjecture on \(\mathrm{cdim}(A)\), the two bounds coincide, which yields a conditional value for \(\mathrm{ed}(G)\).
scientific article; zbMATH DE number 5990157
scientific article