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 \(T_F\) over \(F\) cuts out exactly the rational points, if either \(F\) is a function field or, if \(F={\mathbb Q}\) and \(T_F\) 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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Notes
In this case \(\infty _F=\emptyset \) and \(S\) can be empty, \(P_{\emptyset }\) is then the identity map.
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Acknowledgments
We would like to thank J.-L. Colliot-Thélène, D. Harari, Y. Liang and F. Voloch for helpful discussions. We would also like to thank Liang-Chung Hsia for drawing our attention to [23]. We thank the referee for a careful reading of the manuscript and for pointing out some inaccuracies. The first named author thanks Capital Normal University, where part of this work was done, for its support. The second named author is supported by the ALGANT program in Université de Bordeaux, MPI for mathematics at Bonn from September–October 2014 and NSFC Grant No. 11031004.
