On the BDP Iwasawa main conjecture for modular forms

TL;DR

This paper proves an integral inclusion in the Iwasawa main conjecture for modular forms over imaginary quadratic fields, leading to results on the vanishing of certain Iwasawa invariants.

Contribution

It improves previous results by establishing the integral inclusion of the main conjecture under specific hypotheses, advancing the understanding of Iwasawa theory for modular forms.

Findings

Integral inclusion of the Iwasawa main conjecture under certain hypotheses

Vanishing of the Iwasawa dd-invariants for anticyclotomic Selmer groups

Extension of Kobayashi--Ota's results to integral settings

Abstract

Let $K$ be an imaginary quadratic field where $p$ splits, $p\geq5$ a prime number and $f$ an eigen-newform of even weight and level $N>3$ that is coprime to $p$. Under the Heegner hypothesis, Kobayashi--Ota showed that one inclusion of the Iwasawa main conjecture of $f$ involving the Bertolini--Darmon--Prasanna $p$-adic $L$-function holds after tensoring by $\mathbb{Q}_p$. Under certain hypotheses, we improve upon Kobayahsi--Ota's result and show that the same inclusion holds integrally. Our result implies the vanishing of the Iwasawa $\mu$-invariants of several anticyclotomic Selmer groups.