# Why are Galois Representations so important in Number theory ?

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.

-
I don't know if there is a single story "how Galois rep's became famous", but usually stuff becomes famous for making your life easier, enabling you to prove stuff, etc. Same for Galois reps. To the second point: the most special property of Galois reps I know of is the Frobenius acting on a vector space, giving you a weight decompositon. There is a lot one can do with this... and I second the advice to study representation theory of finite groups. –  Konrad Voelkel Aug 3 '12 at 13:03
I'm not a specialist at all, but here is my (small) understanding. The importance of absolute Galois group in number theory is obvious. But since we can only name some subgroups or cosets up to conjugacy, the only way to study the group is via its representations (mostly linear, but not necessarily). –  BS. Aug 3 '12 at 14:03
But then isn't the reason that we even know those subgroups or cosets up to conjugacy because we know the characters of the Galois group (i.e., 1-dimensional Galois representations, i.e., class field theory) so well? –  stankewicz Aug 3 '12 at 16:22
Thanks quid for your edit. Thank you all for your suggestions. –  Shanmukha_Srinivasan Aug 3 '12 at 16:46
Dear MO, I have a small doubt. Does editing a question brings a down vote ? I got 5 up votes and as soon as I edited, the up-vote count became 3. So I thought that 2 up votes are used up in editing a question . Is that the case ? I am a new user, I don't know about MO completely. Someone please inform me. –  Shanmukha_Srinivasan Aug 3 '12 at 17:16

## 1 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.

-
Good answer, +1. But not complete, let me wait for some of my learned contemporaries to answer, before accepting your answer. –  Shanmukha_Srinivasan Aug 3 '12 at 16:54