On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in $p$-adic Lie Extensions

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