Mladen Dimitrov

Hida families for GL(2) over totally real fields

In the spirit of Iwasawa theory, it is natural and has proved fruitful to study automorphic forms in families. In this general survey we will introduce the main actors in the contruction of families of ordinary Hilbert modular forms, namely towers of quaternionic Shimura varieties and the Hecke operators acting on their cohomology. To transfer information from the family to a member of it, one has to establish Control Theorems for the cohomology of the tower. We will recall classical theorems proved by Hida and mention some more recent results on the subject.