Very strong approximation for certain algebraic varieties

Abstract : Let F be a global field. In this work, we show that the Brauer-Manin condition on adelic points for subvarieties of a torus T over F cuts out exactly the rational points, if either F is a function field or, if F is the field of rational numbers and T is split. As an application, we prove a conjecture of Harari-Voloch over global function fields which states, roughly speaking, that on any rational hyperbolic curve, the local integral points with the Brauer-Manin condition are the global integral points. Finally we prove for tori over number fields a theorem of Stoll on adelic points of zero-dimensional subvarieties in abelian varieties.
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Contributor : Qing Liu <>
Submitted on : Monday, January 18, 2016 - 2:12:41 PM
Last modification on : Wednesday, September 26, 2018 - 11:06:09 AM

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  • HAL Id : hal-01257924, version 1
  • ARXIV : 1405.1988



Qing Liu, Fei Xu. Very strong approximation for certain algebraic varieties. Mathematische Annalen, Springer Verlag, 2015, 363 (3), pp.701-731. ⟨⟩. ⟨hal-01257924⟩



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