  Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has
good supersingular reduction. Let $F_1$ denote the characteristic power series
of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic
$\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest
common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's
sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether
$a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and
$F_2$ in the Iwasawa algebra. This gives new insights into problems posed by
Greenberg and Pollack--Kurihara on these elements.
