From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have Galois Representation as an ingredient !!
I would be very happy listening to :
What made Galois representations so famous ? ( especially in number theory ), I was wondering, may be Galois representations are having some special symmetries that can facilitate the problem solving more easily.
What are the special properties of Galois representations ?
Case Study :
To describe the application of Galois representation in a beautiful manner, I came accross a paper of Skinner and Urban , where they relate the ranks of Selmer Groups to the non-vanishing of $L$-functions. They use this Galois representations as a major ingredient, but due to extensive use of Algebraic Geometry , I was not able to understand the quintessence of the paper. It was so difficult to read. But on the other hand, I know how can one relate the volumes of lattices ( groups ) to the $L$-functions, using Siegel's formula. But I didn't come across any such track in that paper ( The word Siegel is not found in that paper ) . May be they have used some other different approach. I would be very happy in listening to that , as an application of Galois Representation.
Any other good applications of Galois Representations are welcomed with high appreciation.
My Background :
I know number theory ( Mass formula and other things ) and rudimentary theory of Elliptic curves.
I thank everyone for sparing your time in answering / reading my questions and other questions at MO, in-spite of your hectic schedule.