Essential dimension of simple algebras with involutions. (Q2893275)

Let \(F\) be a field, \(\text{Set}\) the category of sets, \(\text{Fields}/F\) the category of field extensions of \(F\) and field \(F\)-homomorphisms, and let \(\mathcal T\colon\text{Fields}/F\to\text{Sets}\) be a functor (called an ``algebraic structure''). For instance, \(\mathcal T(E)\) with \(E\in\text{Fields}/F\) can be the sets \(\text{Alg}_{n,m}\) of isomorphism classes of central simple \(E\)-algebras of degree \(n\) and exponent \(m\) (for an arbitrary fixed divisor \(m\) of \(n\)), torsors (principal homogeneous spaces) over \(E\) under a given algebraic group, etc. For fields \(E,E'\in\text{Fields}/F\), a field homomorphism \(f\colon E\to E'\) over \(F\) and \(\alpha\in\mathcal T(E)\), we write \(\alpha_{E'}\) for the image of \(\alpha\) under the morphism \(\mathcal T(f)\colon\mathcal T(E)\to\mathcal{T}(E')\). An element \(\alpha\in\mathcal T(E)\) is said to be defined over an intermediate field \(K\) of \(E/F\) (and \(K\) is called a field of definition of \(\alpha\)), if there exists \(\beta\in\mathcal T(K)\), such that \(\beta_E=\alpha\), where \(f\) is the natural embedding of \(K\) into \(E\). The essential dimension \(\text{ed}(\alpha)\) of an algebraic structure is defined to be the minimum of transcendence degrees \(\text{trd}(K/F)\), taken over the fields of definition \(K\) of \(\alpha\). For each prime number \(p\), the essential \(p\)-dimension \(\text{ed}_p(\alpha)\) is defined as \(\min\{\text{ed}(\alpha_L)\}\), where \(L\) ranges over all field extensions of \(E\) of degree prime to \(p\).NEWLINENEWLINE The paper under review computes upper bounds for \(\text{ed}(\text{Alg}_{n,2})\) and \(\text{ed}_2(\text{Alg}_{n,2})\). It continues the research in this direction carried out by \textit{A. S. Merkurjev} and the author [in Acta Math. 209, No. 1, 1-27 (2012; Zbl 1258.16023)]. Its starting point are the following results of the quoted joint paper: (i) \(\text{ed}(\text{Alg}_{n,2})=\text{ed}(\text{Alg}_{2^r,2})\) and \(\text{ed}_2(\text{Alg}_{n,2})=\text{ed}_2(\text{Alg}_{2^r,2})\), where \(2^r\) is the largest power of \(2\) dividing \(n\); (ii) \(\text{ed}(\text{Alg}_{4,2})=\text{ed}_2(\text{Alg}_{4,2})=4\) and \(\text{ed}(\text{Alg}_{8,2})=\text{ed}_2(\text{Alg}_{8,2})=8\), provided that \(\text{char}(F)\neq 2\) (for the case of \(\text{char}(F)=2\), see the author's paper [C. R., Math., Acad. Sci. Paris 349, No. 7-8, 375-378 (2011; Zbl 1237.16020)].NEWLINENEWLINE When \(\text{char}(F)\neq 2\) and \(n=2^r\geq 8\), the paper under review shows that \(\text{ed}(\text{Alg}_{n,2})\leq (n-1)(n-2)/2\). Concerning \(\text{ed}_2(\text{Alg}_{n,2})\), it proves that if \(n=2^r\geq 8\), then \(\text{ed}_2(\text{Alg}_{n,2})\leq n^2/4\) in case \(\text{char}(F)=2\), and \(\text{ed}_2(\text{Alg}_{n,2})\leq n^2/16+n/2\), provided that \(\text{char}(F)\neq 2\). In addition, the author obtains that \(\text{ed}_2(\text{Alg}_{16,2})=24\) whenever \(\text{char}(F)\neq 2\). The proof relies on a natural interpretation of \(\text{Alg}_{n,2}\) in terms of nonabelian Galois cohomology.