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Lois de réciprocité supérieures et points rationnels


Authors: J.-L. Colliot-Thélène, R. Parimala and V. Suresh
Journal: Trans. Amer. Math. Soc. 368 (2016), 4219-4255
MSC (2010): Primary 14H25; Secondary 11E72
DOI: https://doi.org/10.1090/tran/6519
Published electronically: September 4, 2015
MathSciNet review: 3453370
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Abstract: Soit $ K=\mathbb{C}((x,y))$ ou $ K=\mathbb{C}((x))(y)$. Soit $ G$ un $ K$-groupe algébrique linéaire connexe. Il a été établi que si $ G$ est $ K$-rationnel, c'est-à-dire de corps des fonctions transcendant pur sur $ K$, si un espace principal homogène sous $ G$ a des points rationnels dans tous les complétés de $ K$ par rapport aux valuations de $ K$, alors il a un point rationnel. Nous montrons ici qu'en général l'hypothèse de $ K$-rationalité ne peut être omise. Nous utilisons pour cela une obstruction d'un nouveau type, fondée sur les lois de réciprocité supérieure sur un schéma de dimension deux. Nous donnons aussi une famille d'espaces principaux homogènes pour laquelle cette obstruction raffinée à l'existence d'un point rationnel est la seule obstruction.


ABSTRACT Let $ K=\mathbb{C}((x,y))$ or $ K=\mathbb{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 transcendental, 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 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.


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Additional Information

J.-L. Colliot-Thélène
Affiliation: C.N.R.S., Université Paris Sud, Mathématiques, Bâtiment 425, 91405 Orsay Cedex, France
Email: jlct@math.u-psud.fr

R. Parimala
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: parimala@mathcs.emory.edu

V. Suresh
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: suresh.venapally@gmail.com

DOI: https://doi.org/10.1090/tran/6519
Received by editor(s): February 1, 2014
Received by editor(s) in revised form: February 8, 2014, and April 15, 2014
Published electronically: September 4, 2015
Article copyright: © Copyright 2015 American Mathematical Society