  Let $f$ be an elliptic modular form and $p$ an odd prime that is coprime to
the level of $f$. We study the link between divisors of the characteristic
ideal of the $p$-primary fine Selmer group of $f$ over the cyclotomic
$\mathbb{Z}_p$ extension of $\mathbb{Q}$ and the greatest common divisor of
signed Selmer groups attached to $f$ defined using the theory of Wach modules.
One of the key ingredients of our proof is a generalization of a result of
Wingberg on the structure of fine Selmer groups of abelian varieties with
supersingular reduction at $p$ to the context of modular forms.
