JournalscmhVol. 91, No. 1pp. 163–202

Motivic construction of cohomological invariants

  • Nikita Semenov

    Ludwig-Maximilians-Universität München, Germany
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Let GG be a group of type E8\mathrm E_8 over Q\mathbb Q such that GRG_\mathbb R is a compact Lie group, let KK be a field of characteristic 0, and

q= ⁣1,1,1,1,1 ⁣q=\langle\!\langle -1,-1,-1,-1,-1\rangle\!\rangle

a 5-fold Pfister form. J.-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem: Is it true that GKG_K is split if and only if qKq_K is hyperbolic?

In the present article we construct a cohomological invariant of degree 5 for groups of type E8\mathrm E_8 with trivial Rost invariant over any field kk of characteristic 0, and putting k=Qk=\mathbb{Q} answer positively this question of Serre. Aside from that, we show that a variety which possesses a special correspondence of Rost is a norm variety.

Cite this article

Nikita Semenov, Motivic construction of cohomological invariants. Comment. Math. Helv. 91 (2016), no. 1, pp. 163–202

DOI 10.4171/CMH/382