In 1972, Serre observed that the Hecke eigenvalues of Eisenstein series can
be p-adically interpolated. In other words, Eisenstein series can be viewed as
a p-adic family parametrized by the weight.

The notion of p-adic variations
of modular forms was later generalized by Hida to include families of ordinary
cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic
space that, loosely speaking, encodes much more general p-adic families of Hecke
eigenforms parametrized by the weight. However, many geometric properties of
the eigencurve are still mysterious. In this talk, we will describe the local nature
of the eigencurve at some particular points corresponding to weight one forms.
We consider weight one Eisenstein series that are irregular at a fixed prime p.
Such forms are not cuspidal in a classical sense, but they become cuspidal when
viewed as p-adic modular forms. Thus, they give rise to points that belong to
the intersection of the Eisenstein locus and the cuspidal locus of the eigencurve.
Following the approach of Bellaiche and Dimitrov in the weight one cuspidal
case, we study this intersection via deformations of Galois representations.