Pseudo-modularity and Iwasawa theory

TL;DR

Under the assumption of Greenberg's conjecture, the paper proves the Gorenstein property of the ordinary eigencurve at specific intersection points and derives new insights into Sharifi's conjecture through deformation rings and $R = ext{T}$ theorems.

Contribution

It establishes the Gorenstein property of the eigencurve at Eisenstein-cuspidal intersections assuming Greenberg's conjecture and connects this to Sharifi's conjecture via deformation theory.

Findings

Proves Gorenstein property of eigencurve at intersection points

Constructs universal ordinary pseudodeformation rings

Derives new results on Sharifi's conjecture

Abstract

We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is Gorenstein at an intersection point between the Eisenstein family and the cuspidal locus. 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.