On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve

TL;DR

This paper investigates the local geometric structure of the eigencurve at weight 1 Eisenstein points, revealing failure of Gorensteinness, and connects it to $p$-adic $L$-functions and classical number theory results.

Contribution

It proves the non-Gorenstein nature of the local ring at certain Eisenstein points and establishes an $R=T$ theorem linking modular forms and Galois representations.

Findings

The eigencurve is étale over the weight space at Eisenstein points.

The local ring at these points is Cohen–Macaulay but not Gorenstein.

The congruence ideal is generated by the Kubota–Leopoldt $p$-adic $L$-function.

Abstract

We prove that the cuspidal eigencurve $C_{\mathrm{cusp}}$ is \'etale over the weight space at any classical weight $1$ Eisenstein point $f$ and meets two Eisenstein components of the eigencurve $C$ transversally at $f$. Further, we prove that the local ring of $C$ at $f$ is Cohen--Macaulay but not Gorenstein and compute the Fourier coefficients of a basis of overconvergent weight $1$ modular forms lying in the same generalised eigenspace as $f$. In addition, we prove an $R=T$ theorem for the local ring at $f$ of the closed subspace of $C$ given by the union of $C_{\mathrm{cusp}}$ and one Eisenstein component and prove unconditionally, via a geometric construction of the residue map, that the corresponding congruence ideal is generated by the Kubota--Leopoldt $p$-adic $L$-function. Finally we obtain a new proof of the Ferrero--Greenberg Theorem and Gross' formula for the derivative of the $p$-adic $L$-function at the trivial zero.