On the Hasse principle for groups of type \(E_ 8\). (English. Russian original) Zbl 0703.20040

Chernousov, V. I.

Sov. Math., Dokl. 39, No. 3, 592-596 (1989); translation from Dokl. Akad. Nauk SSSR 306, No. 5, 1059-1063 (1989).

Let G be a semisimple simply connected group defined over an algebraic number field K. It was a conjecture of J.-P. Serre that the Hasse- Principle holds for such algebraic groups G, i.e. that the natural map \[ w: H^ 1(K,G)\to \prod_{v}H^ 1(K_ v,G) \] is injective. In the case of an orthogonal group \(G=O(f)\) this injectivity of w is equivalent to the classical Hasse-Principle for quadratic forms, while for instance for \(G=PSL\) the projective linear group the Hasse-Principle is equivalent to the classical Hasse-Brauer-Noether-Theorem. The Hasse-Principle was known for all simple, simply-connected semi-simple groups G except for groups of type \(E_ 8\) due to the work of M. Kneser (1962) and G. Harder (1965, 1966). In the paper under review the long standing open question is now answered also for the remaining case \(E_ 8\). Up to now there does not exist a unified proof which works for all the different types simultaneously, the now complete proof consists of a case by case analysis by different ideas for the different types, in which the case \(E_ 8\) played the most complicated role.

Reviewer: H.-J.Bartels

MSC:

20G30	Linear algebraic groups over global fields and their integers
20G10	Cohomology theory for linear algebraic groups
11E57	Classical groups
11E72	Galois cohomology of linear algebraic groups

Keywords:

semisimple simply connected group; algebraic number field; algebraic groups; Hasse-Principle; groups of type \(E_ 8\)