Hasse Principle For Algebraic Groups
The Hasse principle for algebraic groups states that if G is a simply-connected algebraic group defined over the global field k then the map from
is injective, where the product is over all places s of k.
The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.
Kneser (1966) and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group E8 which was only completed by Chernousov (1989) many years after the other cases.
The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.
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