  Let $A$ be an abelian variety defined over a number field $F$ with
supersingular reduction at all primes of $F$ above $p$. We establish an
equivalence between the weak Leopoldt conjecture and the expected value of the
corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not
neccesasily containing the cyclotomic $\Zp$-extension). As an application, we
obtain the exactness of the defining sequence of the Selmer group. In the event
that the $p$-adic Lie extension is one-dimensional, we show that the dual
Selmer group has no nontrivial finite submodules. Finally, we show that the
aforementioned conclusions carry over to the Selmer group of a non-ordinary
cuspidal modular form.
