  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.
