I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting perspective that I wasn't aware of before: that a large chunk of number theory can be thought of as a quest to understand $G = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. For example, part of the reason to study elliptic curves is to describe two-dimensional representations of $G$, and reciprocity laws are secretly about ways to describe the traces of Frobenius elements in various representations. (That's awesome! Why didn't anybody tell me that before?)

Are there number theory textbooks (presumably not introductory, but hopefully not too sophisticated either) which explicitly take this as a guiding principle? I think this is a great idea to organize things like quadratic reciprocity around and I'm wondering if anybody has decided to actually do that at the undergraduate (or introductory graduate, maybe) level.

Edit: In response to some comments and at least one downvote, most of the other questions on MO about the absolute Galois group that I can find are about the state of the art, and the references in them seem fairly sophisticated. But it seems to me there are still interesting things to say along the lines of Fearless Symmetry, but directed to an undergraduate or introductory graduate-level audience as a kind of "second course in number theory." I'm imagining a textbook like Serre's Course in Arithmetic.

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    $\begingroup$ I would say that part of the reason to understand two-dimensional representations is to understand elliptic curves. By the way, several of the number theory questions on MO are on this topic, e.g. mathoverflow.net/questions/11747 and mathoverflow.net/questions/2791 . $\endgroup$
    – Emerton
    Feb 12, 2010 at 4:15
  • $\begingroup$ I also just finished that same book and am curious about this. Suddenly, all those reciprocity laws that I "learned" as an undergrad make sense as part of a unified whole, and that book brought me around to wanting to learn some number theory. $\endgroup$ Feb 12, 2010 at 4:21
  • $\begingroup$ Dear Qiaochu, personally, I would think talking (in person) to a professor or a graduate student (probably even better than to a professor) would actually be more helpful on this aspect. But I bet you already do this, the place where you stay does not lack wonderful graduate students. :) And for this question, as far as I know, no book would really meet your ideal, since as you can see, there are too many things about the absolute Galois group, everything about number theory is about that group. But I think you probably want to learn more about Galois representation. $\endgroup$
    – natura
    Feb 12, 2010 at 4:22
  • $\begingroup$ then you can try Serre's "Abelian l-adic Representations and Elliptic Curves" and Fontaine & Ouyang's "Theory of p-adic Galois representations". They are both pretty good books. $\endgroup$
    – natura
    Feb 12, 2010 at 4:24
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    $\begingroup$ Dear Qiaochu, If you have read Course in Arithmetic, especially the section on modular forms, then I suggest that you read Serre's Bourbaki seminar on modular forms mod p, and Swinnerton-Dyer's Antwerp article on the same topic. They are two wonderful articles, and you will learn much more (from two absolute masters) than you will from any textbook. $\endgroup$
    – Emerton
    Feb 12, 2010 at 6:29

2 Answers 2


If you want to go further in understanding this point of view, I would advise you to begin learning class field theory. It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abelian reciprocity laws, it in fact plays a crucial role in the study of non-abelian reciprocity laws as well.

The texts:

Ireland and Rosen for basic algebraic number theory, a Galois-theoretic proof of quadratic reciprocity, and other assorted attractions.

Cox's book on primes of the form x^2 + n y^2 for an indication of what some of the content of class field theory is in elementary terms, via many wonderful examples.

Serre's Local Fields for learning the Galois theory of local fields

Cassels and Frolich for learning global class field theory

The standard book at the graduate level to learn the arithmetic of non-abelian (at least 2-dimensional) reciprocity laws is Modular forms and Fermat's Last Theorem, a textbook on the proof by Taylor and Wiles of FLT. But it is at a higher level again.

I don't think that you will find a single text on this topic at a basic level (if basic means Course in arithmetic or Ireland and Rosen), because there is not much to say beyond what you stated in your question without getting into the theory of elliptic curves and/or the theory of modular forms and/or a serious discussion of class field theory.

Also, as basic suggests, you could talk to the grad students in your town, if not at your institution, then at the other one down the Charles river, which as you probably know is currently the world centre for research on non-abelian reciprocity laws (maybe shared with Paris). Certainly there are grad courses offered on this topic there on a regular basis.

  • $\begingroup$ Your second to last paragraph suggests that maybe what I requested was too much to ask for at this level. I'll try to keep this in mind. Thanks for the references! $\endgroup$ Feb 12, 2010 at 5:04
  • $\begingroup$ You might also want to learn about elliptic curves. At the broad level we are discussing, they are the same subject, and one has Silverman--Tate, as well a Knapp, Koblitz, and maybe other books, at a more basic level, and then Silverman at the grad level. The modularity theorem for elliptic curves is an incredible 2-dimensional reciprocity law, and understanding at least the statement of the theorem would be a good goal. $\endgroup$
    – Emerton
    Feb 12, 2010 at 5:19
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    $\begingroup$ Diamond and Shurman's book 'A First Course in Modular Forms' takes the approach that Emerton suggested: it attempts to explain the various statements of the modularity theorem. I don't have the book in front of me right now, but I know that one of the later chapters covers Galois representations. $\endgroup$
    – user1073
    Feb 12, 2010 at 5:27
  • $\begingroup$ Diamond and Shurman happens to be available for download on SpringerLink, if anyone at a college with access wants to follow this advice. I was planning on looking at it myself at some point this semester as part of a seminar on modular forms. $\endgroup$ Feb 12, 2010 at 5:40
  • $\begingroup$ @Qiaochu - If you'll be learning about modular forms, you may want to check out the publication section of Ken Ribet's website. He has a number of wonderful survey articles involving Galois reps and their connections to elliptic curves/modular forms. I would particularly recommend the first few sections of "Galois representations attached to eigenforms with nebentypus". $\endgroup$
    – user1073
    Feb 12, 2010 at 5:49

In the 1-dimensional case we are talking about class field theory, and I can't recall having seen anything better than Cox's book (already mentioned by Emerton) in this direction (undergrad etc.). With Artin's reciprocity law (the central role of the Frobenius elements in connection with reciprocity laws wasn't exactly kept secret 8-) under your belt you might want to check out

  • G. Shimura, A reciprocity in law in non-solvable extensions, J. Reine Angew. Math. 221, 209-220 (1966)


  • J. Velu, Lois de reciprocite liees aux courbes elliptiques, Semin. Delange-Pisot-Poitou, 14e annee 1972/73, Fasc. 1, 2, Expose 9, 5 p. (1973)

both of which are rather nice to read, and then continue with the articles by Serre and Swinnerton-Dyer that Emerton suggested (actually they were motivated by concrete problems concerning Ramanujan's $\tau$-function, and they've written quite a bit about it at the time).

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    $\begingroup$ The referenced article of Shimura is a watershed paper; highly seconded! Also, there is an article by Wyman titled "What is a reciprocity law" that I find very helpful when first learning this area; it is in part a commentary on Shimura's article. $\endgroup$
    – Emerton
    Feb 12, 2010 at 12:59

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