# Introductory text on Galois representations

Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" and "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques".

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I think this should be community wiki. Textbook-recommendation questions are the classic example of questions which should be CW. – David White Oct 5 '11 at 22:14
Have you looked at "The Arithmetic of Elliptic Curves" by Joe Silverman? – Kevin Ventullo Oct 6 '11 at 3:53

## 2 Answers

Kevin Ventullo's suggestion of Silverman's book is a very good one. The first examples of Galois representations in nature are Tate modules of elliptic curves, and if you haven't read about them in Silverman's book, you should.

If you have read Silverman's book, a nice paper to read is Serre and Tate's "On the good reduction of abelian varieties". It is a research paper, not a text-book, and is at a higher level than Silverman (especially in its use of algebraic geometry), but it has the merit of being short and beautifully written, and uses Galois representation techniques throughout.

One fantastic paper is Swinnerton-Dyer's article in Lecture Notes 350. Here he explains various things about the Galois representations attached to modular forms. The existence of the Galois representations is taken as a black box, but he explains the Galois theoretic significance of various congruences on the coefficients of the modular forms. Reading it is a good way to get a concrete feeling of what Galois representations are and how you can think about and argue with them.

Another source is Ken Ribet's article "Galois representations attached to modular forms with nebentypus" (or something like that) in one of the later Antwerp volumes. It presupposes some understanding of modular forms, but this would be wise to obtain anyway if you want to learn about elliptic curves, and again demonstrates lots of Galois representation techniques. It would be a good sequel to Swinnerton-Dyer's article.

Yet another good article to read is Ribet's "Converse to Herbrand's criterion" article, which is a real classic. It is reasonably accessible if you know class field theory, know a little bit about Jacobians (or are willing to take some results on faith, using your knowledge of elliptic curves as an intuitive guide), and something about modular forms. Mazur recently wrote a very nice article surveying Ribet's, available here on his web-site.

One problem with reading Serre is that he uses $p$-adic Hodge theory in a strong way, but his language is a bit old-fashioned and out-dated (he was writing at a time when this theory was in its infancy); what he calls "locally algebraic" representations would now be called Hodge--Tate representations. To learn the modern formulation of and perspective on $p$-adic Hodge theory you can look at Laurent Berger's various exposes, available on his web-site. (This will tell you much more than you need to know for Serre's book, though.)

For a two page introduction to Galois representation theory, you could read Mark Kisin's What is ... a Galois representation? for a two-page introduction.

Yet another source is the Fermat's Last Theorem book (Cornell--Silverman--Stevens), which has many articles related to Galois representations, some more accessible than others.

The article of Taylor that Chandan mentioned in a comment is also very nice, although it moves at a fairly rapid clip if you haven't seen any of it before.

Serre's article in Duke 54, in which he explains his conjecture about the modularity of 2-dimensional mod p Galois representations, is also very beautiful, and involves various concrete computations which could be helpful

One last remark: if you do want to understand Galois representations, you will need to have a good understanding of the structure of the Galois groups of local fields (as described e.g. in Serre's book "Local fields"), in particular the role of the Frobenius element, of the inertia subgroup, and of the significance of tame and wild inertia.

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Really great answer ! – Joël Oct 6 '11 at 13:09
yes - thanks very much – Adam Harris Oct 6 '11 at 17:57

There is a very nice introduction to Galois representations in chapter 9 of Diamond and Shurman's book "A First Course in Modular Forms". This is really thorough, e.g. it carefully explains the definition of the group $\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$, decomposition groups, Frobenius elements and so on. There are some exercises too I think. So that might be some use, I hope, although I fear it might be too low-level -- there is still a bit of a gap between it and the two works of Serre you mention.

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David, your advisor's advisor has written a very nice survey article which is available on his website (presently at IAS Princeton). The shorter version has appeared in  mathunion.org/ICM/ICM2002.1/Main/icm2002.1.0449.0474.ocr.pdf  and the longer version in  archive.numdam.org/article/AFST_2004_6_13_1_73_0.pdf  There is also a book by Fontaine and Ouyang which should be available somewhere on the web. – Chandan Singh Dalawat Oct 6 '11 at 3:41
@Chandan: Why are your comments breaking? – S.C. Oct 6 '11 at 6:49
@Chandan: a draft of Fontaine and Ouyang's book can be found here: staff.ustc.edu.cn/~yiouyang/research.html there is also a version on Fontaine's webpage, but I believe it is not as recent. – Laurent Berger Oct 6 '11 at 7:14
@Chandrasekhar: Because I inserted some double dollars here and there to display the links on a line by themselves. – Chandan Singh Dalawat Oct 6 '11 at 8:27