We elucidate the structure of various exceptional subsets appearing in parts I and II in order to prove new results on the Zilber–Pink conjecture for abelian varieties. In particular, we obtain boundedness of height on the intersection of interest for all non-degenerate varieties (in a precise sense). The main idea to prove that our exceptional subsets are closed comes from a result of Bombieri, Masser and Zannier on tori and we follow the same approach through so-called anomalous subvarieties but we have to allow extra generality in the definition. We also use in a crucial way a theorem of Ax on analytic subgroups of algebraic groups.
Cite this article
Gaël Rémond, Intersection de sous-groupes et de sous-variétés III. Comment. Math. Helv. 84 (2009), no. 4, pp. 835–863DOI 10.4171/CMH/183