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Espaces adéliques quadratiques

Published online by Cambridge University Press:  29 June 2016

ÉRIC GAUDRON  and
GAËL RÉMOND
ÉRIC GAUDRON
Affiliation:
Laboratoire de mathématiques, Université Blaise Pascal, UMR 6620 3 place Vasarely, 63178 Aubière Cedex, France. e-mail: Eric.Gaudron@univ-bpclermont.fr
GAËL RÉMOND
Affiliation:
Institut Fourier, Université Grenoble Alpes, UMR 5582 CS 40700, 38058 Grenoble cedex 9, France. e-mail: Gael.Remond@univ-grenoble-alpes.fr
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Abstract

We study quadratic forms defined on an adelic vector space over an algebraic extension K of the rationals. Under the sole condition that a Siegel lemma holds over K, we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 2 , March 2017 , pp. 211 - 247
Copyright
Copyright © Cambridge Philosophical Society 2016 

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