This work has been strongly influenced by the work of M. Kontsevich and Y. Tschinkel. The 2001 colloquium talk of Kontsevich at the Institut de Mathématiques de Jussieu served as a guiding source for us. We are also grateful to him for showing to us the unpublished preprint [KT00]. We further thank A. Thuillier for several interesting discussions, and J.-L. Colliot-Thélène for his help with Lemma A.5.

Our work was carried out at several institutions including the IHES, the École Polytechnique, and the University of Michigan. We gratefully acknowledge their support. The second author was partially supported by the ANR-grant BERKO. The third author was partially supported by the CNRS and the NSF.