In this talk, I will discuss some of the remarkable padic continuity
properties of automorphic forms. For classical modular forms, it was
noted early on that there appear to exist padic families of Hecke
eigenforms  qexpansions with coefficients that are formal power
series in a variable k, which if evaluated at (some) positive integers k
converge padically to the qexpansion of a Hecke eigenform.
In the 1990s Coleman and Mazur showed that all modular forms of finite
slope may be interpolated in this way, and constructed a padic rigid
space, the eigencurve, which can be regarded as a parameter space for
such families. I shall give an introduction to the ColemanMazur theory,
and describe some current research on analogous results for automorphic
forms on higherrank groups.
