##
**Iwasawa theory – past and present.**
*(English)*
Zbl 0998.11054

Miyake, Katsuya (ed.), Class field theory - its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3-12, 1998. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 30, 335-385 (2001).

As the manual for reviewers tells us, the main point of a review is to help the reader decide whether to read the paper under review. This principal mission might be accomplished very briefly here by saying: anybody with just the slightest interest in number theory of whatever kind should be very much encouraged to read this beautiful survey article. Above and beyond this summary statement, the reviewer will at least try to highlight some of the contents of the article.

The author begins with a gentle introduction to the classical theme of Iwasawa theory: \({\mathbb Z}_p\)-extensions, the growth of the \(p\)-part of the class number \(h_n\) of the \(n\)-th field in the tower, and the algebraic constructions (Iwasawa modules) that permit to prove for example the classical result that for all large \(n\), the \(p\)-valuation of \(h_n\) is given by \(\lambda n + \mu p^n + \nu\) for integers \(\lambda,\nu,\mu\), which of course depend on the given \({\mathbb Z}_p\)-extension and among which only \(\nu\) can possibly be negative. As in the whole article, we find a very agreeable mixture of historical and mathematical detail, and the author gives us the simplest and most polished arguments for standard facts. We then learn about Spiegelung, and Iwasawa series (\(p\)-adic L-functions) enter the picture. Before coming to the main conjecture, the author discusses vanishing phenomena. The conjecture that \(\mu\) always vanishes for the cyclotomic \({\mathbb Z}_p\)-extension is presented, and Greenberg gives some hints about the proof of this conjecture by Ferrero and Washington in the absolutely abelian case. Then he turns to the conjecture which bears his name that \(\lambda\) is zero in the cyclotomic case if the base field is totally real. (Cautious people have called this Greenberg’s Question, but it seems that we are now authorized to call it Greenberg’s Conjecture.) Then the author presents, with some preliminary steps, the Main Conjecture which says that the characteristic ideals of certain “cyclotomic Iwasawa modules” are determined by \(p\)-adic \(L\)-functions. The deep results of Mazur-Wiles and Wiles on the Main Conjecture are discussed a little later (in Section 6); there the author explains very nicely why it is natural to look at modular forms and Jacobians if one wants to construct unramified abelian extensions with prescribed Galois properties.

In Section 5 the article turns to elliptic curves. The classical Iwasawa module is now replaced by the Pontryagin dual of the \(p\)-Selmer group of the elliptic curve. There is also a replacement for the \(p\)-adic \(L\)-function, and this leads to Mazur’s Main Conjecture on Selmer groups. Greenberg explains the relation with the Birch-Swinnerton-Dyer conjecture, and indicates results by Rubin and Kato concerning this Main Conjecture. In the CM case one may consider quotient modules of local units modulo elliptic units, and this leads to the one- and two-variable main conjectures of Coates and Wiles, and Yager respectively.

The last two sections (7 and 8) are devoted to Main Conjectures attached to rather general \(p\)-adic \(L\)-functions. Section 7 focusses on the so-called ordinary case: Greenberg discusses here very important developments due to himself, and the last section briefly presents progress in the non-ordinary case by Perrin-Riou, and finally a link between algebraic \(K\)-theory and Iwasawa theory. The reviewer will not attempt to summarize the very important and substantial seventh section, hoping again that readers will look at it, and not at the review.

Iwasawa theory remains a very active area to this day. To illustrate this, let us mention, without pretending in the least that the sample is exhaustive, a few developments that happened after the article under review was written (authors are listed in alphabetical order): there are new results of Greenberg, Iovita, Kobayashi, Pollack and Rubin on the subject of Section 8; Burns, Kurihara, Nguyen Quang Do, Ritter, Weiss and the reviewer developed “equivariant” Iwasawa theory; finally Coates, Schneider, Sujatha and Venjakob made very significant progress on the algebraic side of Iwasawa theory in the noncommutative case.

In conclusion: This is a superb, excellently written survey article by a very prominent expert.

For the entire collection see [Zbl 0968.00031].

The author begins with a gentle introduction to the classical theme of Iwasawa theory: \({\mathbb Z}_p\)-extensions, the growth of the \(p\)-part of the class number \(h_n\) of the \(n\)-th field in the tower, and the algebraic constructions (Iwasawa modules) that permit to prove for example the classical result that for all large \(n\), the \(p\)-valuation of \(h_n\) is given by \(\lambda n + \mu p^n + \nu\) for integers \(\lambda,\nu,\mu\), which of course depend on the given \({\mathbb Z}_p\)-extension and among which only \(\nu\) can possibly be negative. As in the whole article, we find a very agreeable mixture of historical and mathematical detail, and the author gives us the simplest and most polished arguments for standard facts. We then learn about Spiegelung, and Iwasawa series (\(p\)-adic L-functions) enter the picture. Before coming to the main conjecture, the author discusses vanishing phenomena. The conjecture that \(\mu\) always vanishes for the cyclotomic \({\mathbb Z}_p\)-extension is presented, and Greenberg gives some hints about the proof of this conjecture by Ferrero and Washington in the absolutely abelian case. Then he turns to the conjecture which bears his name that \(\lambda\) is zero in the cyclotomic case if the base field is totally real. (Cautious people have called this Greenberg’s Question, but it seems that we are now authorized to call it Greenberg’s Conjecture.) Then the author presents, with some preliminary steps, the Main Conjecture which says that the characteristic ideals of certain “cyclotomic Iwasawa modules” are determined by \(p\)-adic \(L\)-functions. The deep results of Mazur-Wiles and Wiles on the Main Conjecture are discussed a little later (in Section 6); there the author explains very nicely why it is natural to look at modular forms and Jacobians if one wants to construct unramified abelian extensions with prescribed Galois properties.

In Section 5 the article turns to elliptic curves. The classical Iwasawa module is now replaced by the Pontryagin dual of the \(p\)-Selmer group of the elliptic curve. There is also a replacement for the \(p\)-adic \(L\)-function, and this leads to Mazur’s Main Conjecture on Selmer groups. Greenberg explains the relation with the Birch-Swinnerton-Dyer conjecture, and indicates results by Rubin and Kato concerning this Main Conjecture. In the CM case one may consider quotient modules of local units modulo elliptic units, and this leads to the one- and two-variable main conjectures of Coates and Wiles, and Yager respectively.

The last two sections (7 and 8) are devoted to Main Conjectures attached to rather general \(p\)-adic \(L\)-functions. Section 7 focusses on the so-called ordinary case: Greenberg discusses here very important developments due to himself, and the last section briefly presents progress in the non-ordinary case by Perrin-Riou, and finally a link between algebraic \(K\)-theory and Iwasawa theory. The reviewer will not attempt to summarize the very important and substantial seventh section, hoping again that readers will look at it, and not at the review.

Iwasawa theory remains a very active area to this day. To illustrate this, let us mention, without pretending in the least that the sample is exhaustive, a few developments that happened after the article under review was written (authors are listed in alphabetical order): there are new results of Greenberg, Iovita, Kobayashi, Pollack and Rubin on the subject of Section 8; Burns, Kurihara, Nguyen Quang Do, Ritter, Weiss and the reviewer developed “equivariant” Iwasawa theory; finally Coates, Schneider, Sujatha and Venjakob made very significant progress on the algebraic side of Iwasawa theory in the noncommutative case.

In conclusion: This is a superb, excellently written survey article by a very prominent expert.

For the entire collection see [Zbl 0968.00031].

Reviewer: Cornelius Greither (Neubiberg)

### MSC:

11R23 | Iwasawa theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11Gxx | Arithmetic algebraic geometry (Diophantine geometry) |