Lois de réciprocité supérieures et points rationnels

Jean-Louis Colliot-Thélène, Raman Parimala, Venapally Suresh

Published 2013-02-10, updated 2015-09-20Version 3

Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a principal homogeneous space of G has a rational point over K as soon as it has one over each completion of K with respect to a discrete valuation. In this paper we show that one cannot in general do without the K-rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme. We also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points.