I wonder what the prerequisites for learning the representation theory of reductive groups over a p-adic field are? Can someone recommend me any book or article for learning this theory if I wanna clearly know what this theory is about and what kind of applications this theory has?

I just know some fundamental concepts, like Part 1 of Serre's "Linear rep'ns of finite groups", of the rep'ns of finite groups and the structure of p-adic fields. Thanks!

• $p$-adic representations of what? Jan 7, 2011 at 13:43
• Please use the "edit" link under the question, and add some material explaining what you want. Once you have done that, please flag for moderator attention so the question can be reopened. Jan 7, 2011 at 14:56
• Serre's "Abelian l-adic Representations and Elliptic Curves" is a classic. Jan 7, 2011 at 15:55
• Some of the comments on your previous (closed) questions apply equally well here, I think Jan 7, 2011 at 17:57
• The question you wrote is ambiguous, because the term "$p$-adic representation" could mean a representation of a (non-$p$-adic) group by automorphisms of a vector space over a $p$-adic field. If Cartier's article is what you want, then you should change your question to something like: What is a book or article for learning about the representation theory of a reductive group over a $p$-adic field? You should also give some information about what you already know about representation theory, so people can decide whether a given book or article is appropriate for your level. Jan 8, 2011 at 17:18

Hello, I think a good first step is to learn the theory of admissible representations of p-adic groups and for this Godement's notes on Jacquet-Langlands theory and then Casselman's unpublished book on p-adic groups (available from his website) are good starting points. A good way to read Casselman's notes is to rewrite the proof of every theorem explicitly for a small non-GL(2) split group, say GL(3) or Sp(4). If you want to see the big picture shape of the theory and how it is connected to Galois representations you can look at the Trieste notes by Prasad and Raghuram (available from Dipendra Prasad's homepage at Tata). And in your first attempt to learn the theory don't worry too much about supercuspidal representations; just treat them like black boxes or elementary particles. One very nice thing about Godement's notes is that the theory is immediately followed by applications to Hecke theory and L functions.

Let me add a couple of more points to address a question by Alex. You don't need much for Godement's notes; you do, however, need to be comfortable with Tate's thesis (p-adic integration, Haar measure, Poisson summation in the adelic setting, etc). I suppose that's the first thing you need to do if you haven't done already:

Tate's original writeup is amazing and much recommended. There is also the lovely book by Ramakrishnan and Valenza, as well as, of course, Bump's book. If you are already familiar with modular forms (and if not, why aren't you? :-) ) then Gelbart's classical book in the Princeton series is a good place to see the connections between the classical theory and the automorphic theory. When I was just starting to learn automorphic forms, I found Gelbart's treatment very nicely therapeutic.

• Thanks! What are the prerequisites for learning Godement's notes on Jacquet-Langlands theory?
– Alex
Jan 9, 2011 at 14:44
• And what about Bump's "Automorphic forms and representations"?
– Alex
Jan 9, 2011 at 14:47
• The representation theory of reductive p-adic groups is pretty self-contained. Start reading (Bump, Cartier, Casselman, DeBacker, Murnaghan, Prasad/Raghuram are fine notes, off the top of my head), and it should be clear if you are missing anything.
– B R
Jan 9, 2011 at 19:24
• But what are the prerequisites for learning Godement's notes on Jacquet-Langlands theory?
– Alex
Jan 10, 2011 at 7:13
• When I was in grad school I knew I needed to learn this stuff so I carried Gelbart's book around in my backpack all the time, but never got around to reading it. Then one day I went to the student clinic with a bad stomachache and was immediately rushed in for an emergency appendectomy. Subsequently I was laid up in the hospital for a week and the only book I had with me was Gelbart. And that is how I finally learned the basics of automorphic representations. The approach was effective but I cannot recommend it to others.
– JSE
Jan 30, 2011 at 17:35

When I learned this material, 10 or so years ago, I read Cartier's article in Corvalis (linked to by Thomas in the comments above), and then Casselman's notes (mentioned by Ramin). My experience has been that all of this material is a little dry and difficult to learn (due to lack of apparent motivation) if it is not coupled with an understanding of its relationship to the global theory of automorphic forms. In turn, I find automorphic forms difficult to think about without having the link to the classical case of modular forms.

Unfortunately, while there is a lot of literature available which explains the relations between the classical theory of modular forms and the more representation-theoretic and more general theory of automorphic forms, I found a lot of it unsatisfying when I was learning this material for the first time. One article that I like is Deligne's expository article in LNM 349, in which he concretely passes from the theory of modular forms on the upper half-plane to the representation-theoretic view-point of functions on $GL_2$ of the adeles. It might at least serve as a helpful supplement to more comprehensive treatises such as those of Gelbart or Bump.

• @Matt: Sounds like you're contemplating what a good book on the subject would look like! Let me know if you need help writing. :-) Jan 21, 2011 at 5:38

(If you mean Galois repr's): I found these introductory texts on p-adic Galois representations very helpfull:

http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf

http://www.umpa.ens-lyon.fr/~lberger/article05/article05.pdf

http://www.umpa.ens-lyon.fr/~lberger/barcelone/BergerBarcelone.pdf