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

We prove that the cuspidal eigencurve $C_{\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{p}}$ 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 $C_{\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{p}}$ 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.