Families of Galois representations

Given a normalized eigencusp form f and a prime p, there exists a continuous two dimensional
p-adic representation of the absolute Galois group of Q associated to f. Hida constructed p-adic
families of ordinary cusp forms. In particular, he proved that the Galois representations attached
to the ordinary cusp forms can be interpolated. In this talk, we will discuss the variation of
arithmetic properties of such Galois representations in families.