Pseudo-modularity and Iwasawa theory

We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is Gorenstein at an intersection point between the Eisenstein family and the cuspidal locus. In fact, we show that it is a plane singularity. As a corollary, we obtain new results on Sharifi's conjecture. This result is achieved by constructing a universal ordinary pseudodeformation ring and proving an $R = \mathbb T$ result.