Classical results by Serre, Momose, Ribet and Papier assert that the
images of Galois representations attached to modular forms are "as
large as possible" in most cases. I explain how their results admit an
analogue for p-adic families of modular forms, and how the images of
the associated Galois representations can be described in terms of
congruences between families of different types.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/246