On the Iwasawa invariants of Kato's zeta elements for modular forms

TL;DR

This paper investigates the Iwasawa invariants related to Kato's zeta elements for modular forms, extending previous work and establishing the propagation of Kato's main conjecture under certain conditions.

Contribution

It generalizes prior results on Iwasawa invariants and proves the propagation of Kato's main conjecture for higher weight modular forms at arbitrary good primes.

Findings

Established propagation of Kato's main conjecture for higher weight modular forms.

Generalized previous results on Iwasawa invariants without $p$-adic $L$-functions.

Discussed applications to $ ext{±}$ and $lat/ atural$-Iwasawa theory.

Abstract

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us simultaneously. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod $p$ non-vanishing of Kato's zeta elements. The application to the $\pm$ and $\sharp/\flat$-Iwasawa theory for modular forms is also discussed.