Families of $p$-adic cusp forms were first introduced by Hida, later leading to the construction of the eigencurve by Coleman and Mazur. Generalizations to reductive groups of higher rank, called eigenvarieties, are rigid analytic spaces providing the correct setup for the study of $p$-adic deformations of automorphic forms. In order to obtain arithmetic applications, such as constructing $p$-adic $L$-functions or proving explicit reciprocity laws for Euler systems, one needs to perform a meaningful limit process requiring to understand the geometry of the eigenvariety at the point corresponding to the $p$-stabilization of the automorphic form we are interested in.

While the geometry of an eigenvariety at points of cohomological weight is well understood thanks to classicality results, the study at classical points which are limit of discrete series (such as weight $1$ Hilbert modular forms or weight $(2,2)$ Siegel modular forms) is much more involved and the smoothness at such points is a crucial input in the proof of many cases of the Bloch–Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou’s Conjecture.

Far more fascinating is the study of the geometry at singular points, especially at those arising as intersection between irreducible components of the eigenvariety, as those are related to trivial zeros of adjoint $p$-adic $L$-functions.

In this talk we will illustrate this philosophy based on ideas of Joël Bellaïche.