If R(E/Q_\infty) is the fine Selmer group and Y(E/Q_\infty) is its dual, then we know that Y(E/Q_\infty) is a finitely generated \Lambda-module and by a theorem of Kato, it is also torsion. My understanding is that it should thus make sense to want to attach a p-adic L function. 

Can we say what this p-adic L function is? Or does it just follow (trivially) from the Iwasawa main conjecture?