Multiplicative reduction and the cyclotomic main conjecture for $\mathrm{GL}_2$

TL;DR

This paper proves the cyclotomic Iwasawa--Greenberg Main Conjecture for a broad class of modular forms with multiplicative reduction at p, extending known results from the good ordinary case using Hida families and Fitting ideals.

Contribution

It extends the cyclotomic main conjecture to multiplicative reduction cases by leveraging Hida families and Fitting ideal techniques, building on prior good ordinary case results.

Findings

Main conjecture holds for multiplicative reduction cases

Extension from good ordinary to multiplicative cases

Uses Hida families and Fitting ideals for proof

Abstract

We show that the cyclotomic Iwasawa--Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at $p$, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument.