Galois cohomology of reductive groups over number fields

Let G be a connected reductive group over a global field F (a number field or a global function field). Let M=\pi_1(G) denote the *algebraic fundamental group* of G, which is a certain finitely generated abelian group endowed with an action of the absolute Galois group Gal(F^s/F). Using and generalizing a result of Tate for tori, we give a closed formula for the Galois cohomology set  H^1(F,G)  in terms of the Galois module M and the Galois cohomology sets H^1(F_v,G) for the *real* places v of F.

Moreover, let T be an F-torus and let M=\pi_1(T)=X_*(T) denote the cocharacter group of T. We give a closed formula for H^2(F,T) in terms of the Galois module M.

This is a joint work with Tasho Kaletha: https://arxiv.org/abs/2303.04120

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062