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Let $A$ be an abelian variety defined over a number field $F$ withsupersingular reduction at all primes of $F$ above $p$. We establish anequivalence between the weak Leopoldt conjecture and the expected value of thecorank of the classical Selmer group of $A$ over a $p$-adic Lie extension (notneccesasily containing the cyclotomic $\Zp$-extension). As an application, weobtain the exactness of the defining sequence of the Selmer group. In the eventthat the $p$-adic Lie extension is one-dimensional, we show that the dualSelmer group has no nontrivial finite submodules. Finally, we show that theaforementioned conclusions carry over to the Selmer group of a non-ordinarycuspidal modular form.

On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions