Submission date: 28 April 2011.

Abstract:

(With an Appendix by Bas Edixhoven)

The context is the following:
i) in a joint project with D. Masser, A. Pillay and U. Zannier [7], we aim at extending to semi-abelian schemes the Masser-Zannier approach [11] to Conjecture 6.2 of R. Pink's preprint [13]; this conjecture also goes under the name “Relative Manin-Mumford”. Inspired by Anand Pillay's suggestion that the semi-constant extensions of [6] may bring trouble, I found a counter-example, which is described in Section 1 below.
ii) at a meeting in Pisa end of March, Bas Edixhoven found a more concrete way of presenting the counter-example, with the additional advantage that the order of the involved torsion points can be controlled in a precise way: this is the topic of the Appendix.
iii) finally, I realized that when rephrased in the context of mixed Shimura varieties, the construction, far from providing a counter-example, actually supports Pinkâs general Conjecture 1.3 of [13]; a sketch of this view-point is given in Section 2.

Mathematics Subject Classification: 11G18 (Primary) 14H40, 14K12 (Secondary)

Keywords and phrases:

Full text arXiv 1104.5178: pdf, ps.