Forms over number fields and weak approximation

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Abstract

Let K be a number field, and let X ⊆ ℙs-1K be a smooth complete intersection defined over K. In this paper, weak approximation is shown to hold for X provided s exceeds some function of the degree and codimension of X. This is a corollary of a more general result about the number of integral points on certain affine varieties in homogeneously expanding regions. This general result is established via a suitable adaptation of the Hardy-Littlewood Circle Method.

Original languageEnglish (US)
Pages (from-to)11-29
Number of pages19
JournalCompositio Mathematica
Volume106
Issue number1
DOIs
StatePublished - Jan 1 1997

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

  • Forms in many variables
  • Hardy-Littlewood method
  • Weak approximation

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