On the p-adic closure of a subgroup of rational points on an Abelian variety

In 2007, B. Poonen (unpublished) studied the p–adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Belläıche asked the same question for the special case of Abelian varieties. These problems are p–adic analogues of a question raised earlier by B. Mazur on the density of rational points for the real topology. For a simple Abelian variety over the field of rational numbers, we show that the actual p–adic rank is at least the third of the expected value. Acknowledgments The author wishes to take this opportunity to thank Jean Fresnel, who introduced him to p-adic transcendence problems long back. This research started thanks to a discussion with Bjorn Poonen in Tucson during the Arizona Winter School in March 2008. Further discussions on this subject with Cristiana Bertolin in Regensburg shortly afterwards were also useful. The motivation to write this paper was renewed by a correspondence with Joël Belläıche early 2010 [3], while the author was visiting the Harish-Chandra Research Institute in Allahabad, where he had fruitful discussions with Chandan Singh Dalawat.