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

Abstract:
This is a joint work with Adel Betina and Alice Pozzi.  We prove that
the cuspidal p-adic eigencurve is etale over the weight space at any
classical weight 1 Eisenstein point f. Further, we show that it meets
transversely at f each of the two Eisenstein components of the
eigencurve C passing through that point. We prove that the local ring
of C at f is Cohen-Macaulay but not Gorenstein and compute the
q-expansions of a basis of overconvergent weight 1 modular forms lying
in the same   generalised eigenspace as f. The congruences between
cuspidal and Eisenstein families yield a new proof of the
Ferrero-Greenberg and Gross-Koblitz theorem on the order of vanishing
of the Kubota-Leopoldt p-adic L-function at the trivial zero s=0; we
also obtain the formula for its leading term proved by Gross via a new
method.