Congruences and the Iwasawa Main Conjecture for modular forms

In the early 2000s, Ralph Greenberg asked whether the Iwasawa Main Conjecture could be proven in a Hida family of nearly-ordinary p -adic eigencuspforms by propagating it from a known case through congruences. Emerton-Pollack-Weston showed that this is indeed possible when the μ -invariant of such a family is trivial. In this article, we show that this is the case without this assumption, and that in fact such a result holds in general over the universal deformation ring of an irreducible residual modular Galois representation.