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Approximation forte en famille

  • Jean-Louis Colliot-Thélène EMAIL logo and David Harari
Published/Copyright: November 14, 2013
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2016 Issue 710

Abstract

Soient k un corps de nombres et X une k-variété affine lisse intègre fibrée au-dessus de la droite affine 𝔸k1. Supposons que toutes les fibres sont géométriquement intègres, et que la fibre générique est un espace homogène sous un groupe semisimple simplement connexe presque simple G/k(t), les stabilisateurs géométriques étant réductifs connexes. Soit v une place de k telle que la fibration X→𝔸k1 admette une section rationnelle sur le complété kv. Supposons en outre que pour presque tout point x∈𝔸1(kv) le kv-groupe Gx est isotrope. Supposons enfin le groupe de Brauer de X réduit à celui de k. Alors l'approximation forte vaut pour X en dehors de la place v. Let k be a number field and X a smooth integral affine variety equipped with a surjective morphism f:X→𝔸k1 to the affine line. Assume that all fibres of f are split, for instance that they are geometrically integral. Assume that the generic fibre of f is a homogeneous space of a simply connected, almost simple, semisimple group G/k(t), and that the geometric stabilizers are connected reductive groups. Let v be a place of k such that the fibration f acquires a rational section over the completion kv at v. Assume moreover that at almost all points in x∈𝔸1(kv) the specialized group Gx is isotropic over kv. If the Brauer group of X is reduced to the Brauer group of k, then strong approximation holds for X away from the place v.

Les auteurs tiennent à remercier chaleureusement Dasheng Wei pour leur avoir signalé une erreur dans la première version de cet article, et les avoir aidés à la corriger.

Received: 2012-12-13
Revised: 2013-9-12
Published Online: 2013-11-14
Published in Print: 2016-1-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications
  3. The asymptotic expansion of a hypergeometric series coming from mirror symmetry
  4. Convergence of metrics under self-dual Weyl tensor and scalar curvature bounds
  5. Bounded sets of sheaves on Kähler manifolds
  6. Quasiconvexity in relatively hyperbolic groups
  7. Twisted Hilbert transforms vs Kakeya sets of directions
  8. Approximation forte en famille
  9. Arbitrarily large residual finiteness growth
  10. Big denominators and analytic normal forms
https://doi.org/10.1515/crelle-2013-0092

Articles in the same Issue

  1. Frontmatter
  2. The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications
  3. The asymptotic expansion of a hypergeometric series coming from mirror symmetry
  4. Convergence of metrics under self-dual Weyl tensor and scalar curvature bounds
  5. Bounded sets of sheaves on Kähler manifolds
  6. Quasiconvexity in relatively hyperbolic groups
  7. Twisted Hilbert transforms vs Kakeya sets of directions
  8. Approximation forte en famille
  9. Arbitrarily large residual finiteness growth
  10. Big denominators and analytic normal forms
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