Abstract: The deformation theory of modular forms is increasingly
attracting many researchers in arithmetic geometry as it has been an
important step in the proof of Fermat's last theorem by Wiles (and Taylor)
and supplied an effective tool for the study of the $p$-adic Birch and
Swinnerton Dyer conjecture in the proof by Skinner-Urban of divisibility
of the characteristic power series of the Selmer group of a rational
elliptic curve by its $p$-adic $L$-function under appropriate assumptions.
I try to give my background motivation of creating the theory and
describe an outline of the theory.
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