Rational points, 0-cycles and S1 spectra

Our motivating question is: how can one construct invariants that distinguish between the existenceof a rational point and the existence of a 0-cycle of degree 1 on some variety X. We look at thisquestion through the lens of motivic homotopy theory. Here one can use an analog of the classicalPostnikov tower to study properties of the ‘S1-stable homotopy type’ of a variety X. For some special X,this breaks up the homotopy type of X into pieces which can be understood as motives, and thus gives invariants which can be used for a study of the motivating question.