Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-18T14:58:54.341Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

RÉFÉRENCES

[Al] Alon, N. Combinatorial Nullstellensatz. Recent trends in combinatorics (Má-traháza, 1995). Combin. Probab. Comput. 8 (1999), 7–29.CrossRefGoogle Scholar
[Ca1] Cassels, J. Bounds for the least solutions of homogeneous quadratic equations. Proc. Camb. Phil. Soc. 51 (1955), 262–264.Google Scholar
[Ca2] Cassels, J. Addendum to the paper “Bounds for the least solutions of homogeneous quadratic equations”. Proc. Camb. Phil. Soc. 52 (1956) 602.Google Scholar
[CFH] Chan, W.K., Fukshansky, L. et Henshaw, G. Small zeros of quadratic forms outside a union of varieties. Trans. Amer. Math. Soc. 366 (2014), 5587–5612.Google Scholar
[Da] Davenport, H. Note on a theorem of Cassels. Proc. Camb. Phil. Soc. 53 (1957), 539–540.Google Scholar
[dSP] de Seguins Pazzis, C. Invitation aux formes quadratiques. Mathématiques en devenir (Calvage & Mounet, 2011).Google Scholar
[Di] Dietmann, R. Small zeros of quadratic forms avoiding a finite number of prescribed hyperplanes. Canad. Math. Bull. 52 (2009), 63–65.CrossRefGoogle Scholar
[F1] Fukshansky, L. Small zeros of quadratic forms with linear conditions. J. Number Theory 108 (2004), 29–43.Google Scholar
[F2] Fukshansky, L. On effective Witt decomposition and the Cartan–Dieudonné theorem. Canad. J. Math. 59 (2007), 1284–1300.Google Scholar
[F3] Fukshansky, L. Small zeros of quadratic forms over ℚ . Int. J. Number Theory 4 (2008), 503–523.Google Scholar
[F4] Fukshansky, L. Heights and quadratic forms: Cassels' theorem and its generalisations. In Diophantine methods, lattices and arithmetic theory of quadratic forms. Contemp. Math. 587 (Amer. Math. Soc. 2013), p. 77–93.Google Scholar
[GR1] Gaudron, É. et Rémond, G. Lemmes de Siegel d'évitement. Acta Arith. 154 (2012), 125–136.Google Scholar
[GR2] Gaudron, É. et Rémond, G. Minima, pentes et algèbre tensorielle. Israel J. Math. 195 (2013), 565–591.Google Scholar
[GR3] Gaudron, É. et Rémond, G. Corps de Siegel. J. Reine Angew. Math., à paraître. 61 pages.Google Scholar
[Ma] Masser, D. How to solve a quadratic equation in rationals. Bull. London Math. Soc. 30 (1998), 24–28.Google Scholar
[ScH] Schlickewei, H.P. Kleine Nullstellen homogener quadratischer Gleichungen. Monatsh. Math. 100 (1985), 35–45.Google Scholar
[SS1] Schlickewei, H.P. et Schmidt, W. Quadratic geometry of numbers. Trans. Amer. Math. Soc. 301 (1987), 679–690.Google Scholar
[SS2] Schlickewei, H.P. et Schmidt, W. Isotrope Unterräume rationaler quadratischer Formen. Math. Z. 201 (1989), 191–208.Google Scholar
[SS3] Schlickewei, H.P. et Schmidt, W. Bounds for zeros of quadratic forms. In Number theory vol II (Budapest, 1987), Colloq. Math. Soc. János Bolyai 51 (1990), 951–964.Google Scholar
[ScW] Schmidt, W. Small zeros of quadratic forms. Trans. Amer. Math. Soc. 291 (1985), 87–102.Google Scholar
[Th] Thue, A. Eine Eigenschaft der Zahlen der Fermatschen Gleichung. Kristiana Vidensk. Skr. 4 (1911), 1–21.Google Scholar
[Va1] Vaaler, J. Small zeros of quadratic forms over number fields. Trans. Amer. Math. Soc. 302 (1987), 281–296.Google Scholar
[Va2] Vaaler, J. Small zeros of quadratic forms over number fields. II. Trans. Amer. Math. Soc. 313 (1989), 671–686.Google Scholar
[Z] Zhang, S. Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8 (1995), 187–221.Google Scholar