2013/02/10 by Colliot-Thélène, Jean-Louis, Parimala, Raman, Suresh, Venapally
#11E72 #11G99 #14F22 #14G99 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1302.2377
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.