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It is possible to talk about theétale homotopy equivalence of rational points on algebraic varieties by using a relative version of theétale homotopy type. We show that over p-adic fields rational points are homotopy equivalent in this sense if and only if they areétale-Brauer equivalent. We also show that over the real field rational points on projective varieties areétale 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.

Let $$X/\mathbb {F}_{q}$$ X / F q be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of $$\text {GL}_2$$ GL 2 -type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over $$\mathbb {C}$$ C , we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on X come from families of abelian varieties. We have the following application of our main result. If one assumes a strong form of Deligne’s (p-adic) companions conjecture from Weil II, then our conjecture for projective varieties reduces to the conjecture for projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their p-divisible groups.