Dear everyone,
Motivation :
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.
Epilogue :
I thank everyone for sparing your time in answering / reading my questions and other questions at MO, in-spite of your hectic schedule.
-Shanmukha.
Best Answer
For abelian number fields, the Dedekind zeta function factors into Dirichlet $L$-functions. Hecke characters are one-dimensional representations. For possible generalizations to non-abelian field extensions, you will require higher dimensional Galois representations and Artin $L$ functions.
(Maass 1949) Maass wave forms of eigenvalue $1/4$ correspond to 2-dimensional Galois representations, see Bump "Autom.reps...." Chapter 1.9. Similar things happen for modular forms of weight one (Hecke 1925). In general, automorphic representations and Galois representations are expected to be in a certain correspondence (the $L$ functions and the root numbers should be the same).
A possible conceptual explanation for the importance of Galois representations delivers the Tannaka-Krein theorem. Roughly, this states that knowing the representation theory is equivalent to knowing the group. The group you want to understand is the absolute Galois groups (with a profinite topology) via its Galois representations, and understand the Galois representation via automorphic forms.
Perhaps one famous example is the Taniyama Shimura conjecture and consequently Fermat's last theorem: A certain construction with the elliptic curve gave a Galois representation, and the later was then shown to correspond to an automorphic form.