Galois cohomology of reductive groups over global fields

Let F be a number field (say, the field of rational numbers Q) or a p-adic field (say, the field of p-adic numbers Q_p), or a global function field (say, the field of rational functions of one variable over a finite field F_q). Let G be a connected reductive group over F (say, SO(n) ). One needs the first Galois cohomology set H^1(F,G) for classification problems in algebraic geometry and linear algebra over F. In the talk, I will give closed formulas for H^1(F,G) when F is as above, in terms of the algebraic fundamental group \pi_1(G) introduced by the speaker in 1998. All terms will be defined and examples will be given.

The talk is based on a joint work with Tasho Kaletha.