Abstract

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 a torus over a global field F and let M=\pi_1(T)=X_{\star}(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.