The study of p-adic deformations of automorphic forms was initiated by Hida in the 1980s, following his discovery of systematic congruences between the Fourier coefficients of modular forms. The eigencurve, introduced by Coleman and Mazur, offers a geometric framework for understanding these congruences, and has since become a central tool in tackling deep number-theoretic conjectures, such as the Birch and Swinnerton-Dyer conjecture. At non-critical classical points of integer weight k>1, the eigencurve is known to be smooth, thanks to the classicality theorems of Hida and Coleman. In contrast, the structure of the eigencurve at weight k=1 is significantly more subtle and intricate. In this talk, after reviewing the seminal work of Bellaïche and Dimitrov in the so-called p-regular case (i.e. when crystalline Frobenius does not act by a scalar), I will present joint work with Adel Betina and Alice Pozzi giving a complete description of the local geometry in the more delicate p-irregular case. Time permitting, I will also discuss ongoing work comparing the completed local rings of the eigencurve at p-irregular weight one points with suitable Galois deformation rings.