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In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroupof rational points on a commutative algebraic group. More recently, J.Bella\"iche asked the same question for the special case of Abelian varieties.These problems are $p$--adic analogues of a question raised earlier by B. Mazuron the density of rational points for the real topology. For a simple Abelianvariety over the field of rational numbers, we show that the actual $p$--adicrank is at least the third of the expected value.

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