# How do we study Iwasawa theory?

What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about? And what are people researching on it?

• Does the Wikipedia entry for "Iwasawa theory" go any way to answering your queries? Nov 27, 2009 at 4:40
• Thank you! wikipedia is a good place, but sometimes, they want it to be more accessible to more people, thus they sometimes don't give a more suitable answer to more involved inquiries. the wiki entry of "Iwasawa theory" is good, but it practically has no explanation about p-adic L-function or the Iwasawa module. It spent too much space to explain the Iwasawa algebra, which could be misleading at some points... just my 2 cents. Nov 27, 2009 at 5:53
• The book Number theory 3 by Kurokawa, Kurihara, Saito, has a nice chapter which introduces the reader to Iwasawa theory. Sep 6, 2018 at 15:05
• Oct 20, 2018 at 10:00

Iwasawa theory has its origins in the following counterintuitive insight of Iwasawa: instead of trying to describe the structure of any particular Galois module, it is often easier to describe every Galois module in an infinite tower of fields at once.

The specific example that Iwasawa studied was the $p$-Sylow subgroup of the class group of $K_n = \mathbb{Q}(\zeta_{p^n})$. It's naturally a $\mathbb{Z}_p$-module as well as a $G_n$ = Gal$(K_n/K_1)$-module, but the group ring $\mathbb{Z}_p[G_n]$ isn't very nice; it's not a domain, for instance. If we instead look at the inverse limits of the $p$ parts of the class groups of all the fields $K_n$ at once, as modules over $\mathbb{Z}_p[G_n]$, we get a module over the inverse limit $\varprojlim\mathbb{Z}_p[G_n]$. This ring is much easier to understand; it's a complete 2-dimensional regular local ring that is (non-canonically) isomorphic to a power series ring, and there is a strong structure theorem for modules over this ring. Using this structure theorem, Iwasawa proved many theorems about the class numbers of cyclotomic fields. For a simple example: $p$ divides the class number of one of the fields $K_n$ if and only if it divides the class number of all of the fields $K_n$.

There's an even bigger payoff to the theory: a profound connection with special values of $L$-functions. In the function field case, Weil had interpreted the Hasse-Weil $L$-function as computing the characteristic polynomial of Frobenius acting on the Jacobian of a curve. Iwasawa's idea was that the analogue for number fields should be the "characteristic ideal" of the ring $\varprojlim\mathbb{Z}_p[G_n]$ acting on ideal class groups. It turns out this characteristic ideal has a generator that is essentially the same as a $p$-adic $L$-function closely related to the ordinary Dirichlet $L$-functions. This was Iwasawa's "main conjecture" and is now a theorem. It implies the Herbrand-Ribet theorem and essentially every classical result relating cyclotomic fields and zeta values.

There have been many generalizations since but it's safe to call an area "Iwasawa theory" if it studies some Galois representation ranging over an infinite tower of fields and connects it to $p$-adic $L$-functions. The most fruitful Galois modules from the point of view of $L$-functions seem to be Bloch and Kato's generalized Selmer groups; the ideal class group can be interpreted as a Selmer group, and so can the classical Selmer group of an abelian variety. There's a lot of current research in this area.

To start reading, I recommend Washington's book on cyclotomic fields. Chapter 13 is fun and is a good use of some of the main techniques of Iwasawa theory. You don't need anything but the basic background in chapters 1-4 to read sections 1-4 of Chapter 13, which contain the types of theorem I was referring to in the first two paragraphs of this answer. The explicit computations in the first ten chapters also give the link to $p$-adic L-functions. If you know some algebraic number theory, you should be fine to read this book. I also strongly recommend Greenberg's PCMI notes on the Iwasawa theory of elliptic curves, which you can find here:

http://www.math.washington.edu/~greenber/Park.ps

If you're comfortable with class field theory, and have read the first few sections of Chapter 13 in Washington, then Coates and Sujatha's recent book, Cyclotomic Fields and Zeta Values, is a pleasure to read.

• tnank you! i think the notes by Greenberg should be great! Did you ever read Kato's work? they seem really advanced. Nov 27, 2009 at 5:55

If I were you I'd read the following first: On $Z_l$-Extensions of Algebraic Number Fields, The Annals of Mathematics, Second Series, Vol. 98, No. 2 (Sep., 1973), pp. 246-326, By Iwasawa.

This THE paper every Iwasawa fan should read.

Then I'd take a look to On $p$-adic $L$-functions and cyclotomic fields Nagoya Math. J. Volume 56 (1975), 61-77, by Ralph Greenberg. Also check the continuation of Greenberg's paper.

The above papers are somehow the foundations of classical Iwasawa theory.

After the above, I highly recommend: Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183–266, by Mazur.

Check out Otmar Venjakob thesis, and his papers, to have an idea of non commutative Iwasawa theory.

About books I'd definitely have in my library Cyclotomic Fields and Zeta Values. Also Ralph Greenberg has been working on a book on Iwasawa theory, and I think you can find some chapters of it in his webpage.

Part 2 of Kato's lectures was never published as I am sure everyone is aware, however, Kato's paper with Fukaya- "A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory" is a replacement (and of course an extension) of part 2.

Kato's lectures on the generalized Iwasawa conjecture and how it fits into the general picture of zeta values, $$p$$-adic periods etc., are intended to be "joyful lectures good for beginners" (unfortunately, the article contains only part I, does anyone know about part II?). Another overview is given in Kato's ICM lecture. Pottharst wrote a short note "What is Iwasawa theory about (in my opinion)?".

• The first and last links no longer work. Does anyone know where to find them? Dec 30, 2020 at 23:23
• New link for Kato’s lectures: link.springer.com/chapter/10.1007%2FBFb0084729 Mar 8, 2021 at 9:53
• The article contains both, Part I and Part II. On pages 52–53, Kato provides a table of contents with chapters and sections, some of them marked with the sign "*)", and says, "Part I (resp. II) consists of the sections without (resp. with) the sign *)." Apr 15, 2023 at 11:28
• @DavidCorwin The links have been fixed now, just FYI. Apr 15, 2023 at 14:08

You can look up Iwasawa's old papers like Analogy between number and function fields which is pretty much the motivation for the subject. Also Iwasawa's book on p-adic L-functions is a hard but good book to read

I collected a list of references for Iwasawa theory here, including links to various surveys which hasn't been mentioned so far.