On fine Selmer groups and signed Selmer groups of elliptic modular forms

TL;DR

This paper investigates the relationship between the divisors of the characteristic ideal of the fine Selmer group of an elliptic modular form and the gcd of signed Selmer groups, extending known results to modular forms.

Contribution

It generalizes Wingberg's results on fine Selmer groups to the setting of modular forms and explores their connection with signed Selmer groups via Wach modules.

Findings

Established a link between characteristic ideal divisors and signed Selmer groups.

Extended Wingberg's structure results to modular forms with supersingular reduction.

Provided new insights into the structure of fine Selmer groups for elliptic modular forms.

Abstract

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.