Étale homotopy equivalence of rational points on algebraic varieties

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It is possible to talk about the étale homotopy equivalence of rational points on algebraic varieties by using a relative version of the étale homotopy type. We show that over p-adic fields rational points are homotopy equivalent in this sense if and only if they are étale-Brauer equivalent. We also show that over the real field rational points on projective varieties are étale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Châtelet surfaces.