If $R(E/\Bbb Q_{\infty})$ is the fine Selmer group and $Y(E/\Bbb Q_{\infty})$ is its dual, then we know that $Y(E/\Bbb Q_{\infty})$ 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?