Lois de r\'eciprocit\'e sup\'erieures et points rationnels

TL;DR

This paper investigates the existence of rational points on principal homogeneous spaces over certain function fields, introducing a new obstruction based on higher reciprocity laws that extends previous results requiring K-rationality.

Contribution

It demonstrates that the K-rationality assumption is essential and introduces a novel obstruction using higher reciprocity laws on 2-dimensional schemes.

Findings

K-rationality is necessary for certain rational point existence results

Introduces a new obstruction based on higher reciprocity laws

Provides examples where the refined obstruction precisely determines rational points

Abstract

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.