In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of 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 classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.
In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.