On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms

TL;DR

This paper introduces a numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato's Euler systems for modular forms, applicable across Hida families without Eisenstein congruences.

Contribution

It presents a new, efficient numerical method for checking the main conjecture and Euler system indivisibility, avoiding Eisenstein congruence techniques.

Findings

Numerical criterion successfully verifies the main conjecture in examples.

Explicit computation of the integral image under the dual exponential map.

Criterion applies uniformly to all members of a Hida family in the ordinary case.

Abstract

We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato's Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary case, the criterion works for all members of a Hida family once and for all. The key ingredient is the explicit computation of the integral image of the derived Kato's Euler systems under the dual exponential map. We provide explicit new examples at the end. This work does not appeal to the Eisenstein congruence method at all.