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Forms over number fields and weak approximation

Published online by Cambridge University Press:  04 December 2007

C. M. SKINNER
C. M. SKINNER
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544 e-mail: cmcls@math.princeton.edu
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Abstract

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Let $K$ be a number field, and let $X \subseteq P^{s-1}_K$ 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.

Keywords

weak approxmiationHardy-Littlewood methodforms in many variables
Type
Research Article
Information
Compositio Mathematica , Volume 106 , Issue 1 , March 1997 , pp. 11 - 29
Copyright
© 1997 Kluwer Academic Publishers