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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Lois de réciprocité supérieures et points rationnels
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by J.-L. Colliot-Thélène, R. Parimala and V. Suresh PDF
Trans. Amer. Math. Soc. 368 (2016), 4219-4255 Request permission

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.

References
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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
  • MR Author ID: 50705
  • Email: jlct@math.u-psud.fr
  • R. Parimala
  • Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
  • MR Author ID: 136195
  • Email: parimala@mathcs.emory.edu
  • V. Suresh
  • Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
  • Email: suresh.venapally@gmail.com
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 4219-4255
  • MSC (2010): Primary 14H25; Secondary 11E72
  • DOI: https://doi.org/10.1090/tran/6519
  • MathSciNet review: 3453370