# Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms

## TL;DR

This paper investigates the behavior of Iwasawa invariants of anticyclotomic Selmer groups for congruent modular forms, revealing relationships between their invariants even when the groups have positive coranks.

## Contribution

It generalizes previous results by establishing explicit relations between invariants of Selmer groups for congruent forms with positive coranks under the Heegner hypothesis.

## Key findings

- Mu-invariant vanishes for one form iff it vanishes for the other.
- Lambda-invariants are related by an explicit formula.
- Results extend known theorems to cases with positive coranks.

## Abstract

We study the variation of Iwasawa invariants of the anticyclotomic Selmer groups of congruent modular forms under the Heegner hypothesis. In particular, we show that even if the Selmer groups we study may have positive coranks, the mu-invariant vanishes for one modular form if and only if it vanishes for the other, and that their lambda-invariants are related by an explicit formula. This generalizes results of Greenberg-Vatsal for the cyclotomic extension, as well as results of Pollack-Weston and Castella-Kim-Longo for the anticyclotomic extension when the Selmer groups in question are cotorsion.