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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!

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Three applications which come to mind:

  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).

  2. Another application is in proving modularity of Galois representations, treating deformation rings of Galois representations of arbitrary weight instead of fixed weight makes certain algebro-geometric constructions feasable, see the papers of Skinner and Wiles for instance.

  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.

On the other hand, the existence of a Hida family means that one can ask some natural questions about the variation of Iwasawa invariants along the family, this is what Emerton Pollack and Weston "Variation of Iwasawa invariants..." do. This helps clarify the nature of such invariants.

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