# Motivations of families of modular forms, elliptic curves and Galois representations?

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!

1. Greenberg and Stevens "p-adic L-functions and p-adic periods of modular forms" use the 2-variable p-adic L-function (associated to a Hida family) to prove a formula for $$L_p'(E,1)$$ for an elliptic curve with split multiplicative reduction in terms of the L-invariant of the elliptic curve. This formula was discovered experimentally by Mazur Tate and Teitelbaum (motivated by p-adic analogs of the BSD).
3. The proof of the Main conjecture of Iwasawa theory for totally real fields (due to Wiles) uses the theory of $$\Lambda$$-adic forms, so can be added to the list.