I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois representation of a modular form or the Tate module of an abelian variety, but I have not considered the families, though I often hear someone mention some words like the hida family or the eigencurve.

So could someone give some motivations or some classic reference? It seems that Serre uses the family of modular forms to define the p-adic modular forms and extend the definition field of p-adic L function, are there some motivations else?

Thanks!