Zero Cycles of Degree One on Principal Homogeneous Spaces
Let k be a field and let G be a connected linear algebraic group over k. Let X be a principal homogeneous space under G over k. Jean-Pierre Serre has asked the following: "If X admits a zero cycle of degree one, does X have a k-rational point?" We give a positive answer to the question in two settings: 1. The field k is of characteristic different from 2 and the group G is simply connected or adjoint and of classical type. 2. The field k is perfect and of virtual cohomological dimension at most 2 and the simply connected group associated to G satisfies a Hasse principle over k.