Vol. 9, No. 4, 2015

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Étale homotopy equivalence of rational points on algebraic varieties

Ambrus Pál

Vol. 9 (2015), No. 4, 815–873

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.

étale homotopy, rational points
Mathematical Subject Classification 2010
Primary: 14F35
Secondary: 14G05
Received: 9 September 2013
Revised: 11 February 2015
Accepted: 11 March 2015
Published: 30 May 2015
Ambrus Pál
Department of Mathematics
Imperial College
180 Queen’s Gate
London SW7 2AZ
United Kingdom