Critical p-adic L-functions for Hilbert modular forms

In this talk I want to discuss the p-adic interpolation of special values of L-functions in the setting of Hilbert modular forms. The interpolating functions, called p-adic L-functions, are generally constructed under a hypothesis of "small slope" and we will explain how to extend the constructions beyond that case. The main object of our study will be the local geometry of a p-adic family of Hilbert modular forms.