Essential dimension of Albert algebras (Q2922857)

As mentioned by the author, the classification question of Albert algebras over arbitrary fields, has many open questions. One of them deals with the number \(d\) of algebraically independent parameters needed to describe Albert algebras up to isomorphism. Fix a base field \(F\). Formally speaking \(d\) is the smallest number \(d\) such that for any Albert algebra over a field extension of \(F\), there exists a field of definition whose degree of transcendence over \(F\) is at least \(d\). This number \(d\) agrees with the essential dimension of \(F_4\) (denoted \(\mathrm {ed}(F_4)\)). It has been proved that \(5\leq\mathrm {ed}(F_4)\). Also an upper bound \(\mathrm {ed}(F_4)\leq 19\) is known for fields of characteristic zero (more details in the work under review). The main contribution of this paper is that if the characteristic of the ground field is different from \(2\) and \(3\), then NEWLINE\[NEWLINE\mathrm {ed}(F_4)\leq 7.NEWLINE\]NEWLINE As a corollary it is proved that for simply connected \(E_6\) one has \(\mathrm {ed}(E_6)\leq 8\).NEWLINENEWLINEThe main tools involved include the split Albert algebra \(J_s\), the nine-dimensional subalgebra \(E_s\) of \(J_s\) (whose entries range in the linear span of \(\{1,i\}\)) and the three dimensional subalgebra \(L_s\) of diagonal matrices. Next, the author defines \(G:=\Aut(J_s,E_s,L_s)\) the subgroup of automorphisms of \(J_s\) that preserve the subalgebras \(E_s\) and \(L_s\). It is proved that, as a group scheme, \(G\cong(((\mathrm{SL}_3\times \mathrm G_m^2)/\mu_3)\rtimes \mathbb Z_2)\rtimes S_3\) (the dimension of \(G\) which is \(10\) as one can see from the previous formula is used in the final computation of the upper bound of \(\mathrm {ed}(F_4)\)). Also, as shown in the paper, the Galois cohomology set \(H^1(F,G)\) is in bijective correspondence with the set of isomorphism classes of forms of \((J_s,E_s,L_s)\). Furthermore, for arbitrary field \(F\), the inclusion \(G\subset F_4\) induces a surjection \(H^1(F,G)\to H^1(F,F_4)\) and this is in the core of the upper bound computation.