Skip to main content
    • Aa
    • Aa

Espaces adéliques quadratiques

  • ÉRIC GAUDRON (a1) and GAËL RÉMOND (a2)

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.

Hide All
[Al] Alon N. Combinatorial Nullstellensatz. Recent trends in combinatorics (Má-traháza, 1995). Combin. Probab. Comput. 8 (1999), 729.
[Ca1] Cassels J. Bounds for the least solutions of homogeneous quadratic equations. Proc. Camb. Phil. Soc. 51 (1955), 262264.
[Ca2] Cassels J. Addendum to the paper “Bounds for the least solutions of homogeneous quadratic equations”. Proc. Camb. Phil. Soc. 52 (1956) 602.
[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), 55875612.
[Da] Davenport H. Note on a theorem of Cassels. Proc. Camb. Phil. Soc. 53 (1957), 539540.
[dSP] de Seguins Pazzis C. Invitation aux formes quadratiques. Mathématiques en devenir (Calvage & Mounet, 2011).
[Di] Dietmann R. Small zeros of quadratic forms avoiding a finite number of prescribed hyperplanes. Canad. Math. Bull. 52 (2009), 6365.
[F1] Fukshansky L. Small zeros of quadratic forms with linear conditions. J. Number Theory 108 (2004), 2943.
[F2] Fukshansky L. On effective Witt decomposition and the Cartan–Dieudonné theorem. Canad. J. Math. 59 (2007), 12841300.
[F3] Fukshansky L. Small zeros of quadratic forms over . Int. J. Number Theory 4 (2008), 503523.
[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. 7793.
[GR1] Gaudron É. et Rémond G. Lemmes de Siegel d'évitement. Acta Arith. 154 (2012), 125136.
[GR2] Gaudron É. et Rémond G. Minima, pentes et algèbre tensorielle. Israel J. Math. 195 (2013), 565591.
[GR3] Gaudron É. et Rémond G. Corps de Siegel. J. Reine Angew. Math., à paraître. 61 pages.
[Ma] Masser D. How to solve a quadratic equation in rationals. Bull. London Math. Soc. 30 (1998), 2428.
[ScH] Schlickewei H.P. Kleine Nullstellen homogener quadratischer Gleichungen. Monatsh. Math. 100 (1985), 3545.
[SS1] Schlickewei H.P. et Schmidt W. Quadratic geometry of numbers. Trans. Amer. Math. Soc. 301 (1987), 679690.
[SS2] Schlickewei H.P. et Schmidt W. Isotrope Unterräume rationaler quadratischer Formen. Math. Z. 201 (1989), 191208.
[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), 951964.
[ScW] Schmidt W. Small zeros of quadratic forms. Trans. Amer. Math. Soc. 291 (1985), 87102.
[Th] Thue A. Eine Eigenschaft der Zahlen der Fermatschen Gleichung. Kristiana Vidensk. Skr. 4 (1911), 121.
[Va1] Vaaler J. Small zeros of quadratic forms over number fields. Trans. Amer. Math. Soc. 302 (1987), 281296.
[Va2] Vaaler J. Small zeros of quadratic forms over number fields. II. Trans. Amer. Math. Soc. 313 (1989), 671686.
[Z] Zhang S. Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8 (1995), 187221.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 114 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd October 2017. This data will be updated every 24 hours.