On fine Selmer groups and signed Selmer groups of 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.

Some minor corrections and updates on references

References in corpus (2)

• On fine Selmer groups and the greatest common divisor of signed and chromatic $p$-adic $L$-functions
• Iwasawa-Greenberg main conjecture for non-ordinary modular forms and Eisenstein congruences on $\mathrm{GU}(3,1)$

Cited by in corpus (3)

• A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes
• On fine Mordell-Weil groups over ${\mathbb{Z}}_p$-extensions of an imaginary quadratic field
• Fine Selmer groups of modular forms