Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-33068-2/hbk). x, 113 p. EUR 53.45 (2006).

The authors’ aim in this book is to present a proof of the so-called Iwasawa Main Conjecture for the $p$th cyclotomic field ${\Bbb Q}(\mu_p)$ ($p$ an odd prime). This conjecture, characterized by the authors as “the deepest result we know about the arithmetic of cyclotomic fields”, was originally proven by {\it B. Mazur} and {\it A. Wiles} [Invent. Math. 76, 179--330 (1984;

Zbl 0545.12005)] who used the theory of modular curves and quite a heavy machinery from algebraic geometry. A simpler proof based on ideas of F. Thaine and V. Kolyvagin was later published by {\it K. Rubin}; see [{\it S. Lang}, Cyclotomic Fields I and II combined 2nd ed., with an appendix by K. Rubin, Graduate Texts in Mathematics, 121. New York etc.: Springer-Verlag (1990;

Zbl 0704.11038)]. The Main Conjecture has its origin in {\it K. Iwasawa}’s article [J. Math. Soc. Japan 16, 42--82 (1964;

Zbl 0125.29207)]. The conjecture provides an intimate link between two very different objects, one purely arithmetic and the other analytic: the former is an ideal of an algebra determined by the cyclotomic ${\Bbb Z}_p$-extension ${\Bbb Q}(\mu_{p^\infty})$ over ${\Bbb Q}(\mu_p)$ and the latter is the $p$-adic zeta function of ${\Bbb Q}(\mu_p)$. The authors remark that the conjecture can be seen as the simplest example of a vast array of subsequent, unproven “main conjectures” in modern arithmetic geometry, the most celebrated example of these being the conjecture of Birch and Swinnerton-Dyer for elliptic curves.
To formulate the Main Conjecture in the form treated in the present text, denote by $F_\infty$ the maximal real subfield of ${\Bbb Q}(\mu_{p^\infty})$ and let $G=\text{{Gal}}(F_\infty/{\Bbb Q})$. Let $M_\infty$ be the maximal abelian $p$-extension of $F_\infty$ unramified outside $p$. The action of $G$ on the group $X_\infty = \text{{Gal}}(M_\infty/F_\infty)$ endowes $X_\infty$ with a structure of a module over the Iwasawa algebra $\Lambda(G)$. Recall that this algebra is defined as the projective limit of ${\Bbb Z}_p[G/H]$, where $H$ runs through the open subgroups of $G$. The module $X_\infty$ is finitely generated and torsion; thus, up to a finite module, it is isomorphic to a direct sum $\bigoplus_{i=1}^r \Lambda(G)/\langle f_i \rangle$ for some nonzero elements $f_i$ of $\Lambda(G)$. The Main Conjecture asserts that the ideal of $\Lambda(G)$ generated by $f_1\cdots f_r$, the “$G$-characteristic ideal” of $X_\infty$, is essentially given by a zeta function $\zeta_p$ which is a $p$-adic analogue of the Riemann zeta function $\zeta$. More precisely, $$ \text{{ch}}_G(X_\infty) = I(G)\zeta_p, $$ where $I(G)$ is the augmentation ideal of $\Lambda(G)$ and $\zeta_p$ is the unique pseudo-metric on $G$ defined by $$ \int_G \chi(g)^k\,d\zeta_p = (1-p^{k-1})\zeta(1-k) \quad (k=2,4,\dots), $$ with $\chi$ the cyclotomic character of ${\Bbb Q}(\mu_{p^\infty})$. After a brief introductory chapter reviewing the Main Conjecture and describing the main points in its subsequent proof, the authors go on with some preliminaries on local units, Iwasawa algebras and $p$-adic measures. A crucial step in the actual proof is Iwasawa’s theorem about the structure of $U_\infty^1$, the group defined by the local principal units (attached to $F_\infty$), modulo its subgroup $C_\infty^1$ of cyclotomic units. This theorem says that, as a $\Lambda(G)$-module, $U_\infty^1/C_\infty^1$ is canonically isomorphic to $\Lambda(G)/I(G)\zeta_p$. The authors give for this result an elementary proof avoiding the use of class field theory. Their proof is adopted from a work by the first author and {\it A. Wiles} [J. Aust. Math. Soc., Ser. A 26, 1--25 (1978;

Zbl 0442.12007)] up to a simplification due to {\it R. Coleman} [Invent. Math. 53, 91--116 (1979;

Zbl 0429.12010)]. The proof of the Main Conjecture is then completed in the spirit of Rubin [op.cit.] by arguments using Euler systems. The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details. According to the authors, the book is intended for graduate students and the non-expert in Iwasawa theory. I think that also the expert may enjoy reading this kind of unified treatment of such a beautiful theme.