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

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In 2007, Poonen (unpublished) studied the p-adic closure of a subgroup of rational points on a commutative algebraic group. More recently, Bellaiche asked the same question for the special case of Abelian varieties. These problems are p-adic analogues of a question raised earlier by 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.