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Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ hasgood supersingular reduction. Let $F_1$ denote the characteristic power seriesof 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 greatestcommon divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung'ssharp 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 byGreenberg and Pollack--Kurihara on these elements.

On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions