Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms -- Part II

TL;DR

This paper investigates the structure of anticyclotomic Selmer groups associated with $p$-ordinary modular forms over imaginary quadratic fields, extending previous results and refining earlier Iwasawa invariant findings.

Contribution

It proves the absence of proper finite index submodules in these Selmer groups and improves earlier results on Iwasawa invariants for congruent modular forms.

Findings

Selmer group has no proper finite index submodules

Generalizes Bertolini's elliptic curve results

Provides corrections and improvements to previous Iwasawa invariant results

Abstract

We study the Selmer group associated to a $p$-ordinary newform $f \in S_{2r}(\Gamma_0(N))$ over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K/\mathbb{Q}$. Under certain assumptions, we prove that this Selmer group has no proper $\Lambda$-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors.