nLab Iwasawa theory

Compared with Kummer’s criterion and class number formula, Iwasawa theory is finer in the point that it describes not only the class number, i.e. the order of the ideal class group, but also the action of the Galois group on the ideal class group. In fact, one could even say that the aim of Iwasawa theory is to describe Galois actions on arithmetic objects in terms of zeta values. [Kato 06]

Idea

Iwasawa theory is the study of certain modules of arithmetic interest over the Iwasawa algebra $\mathbb{Z}_{p}[[\Gamma]]$, where $\Gamma$ is the subgroup of the Galois group $\mathrm{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})\cong (1+p\mathbb{Z}_{p})\times\mu_{p-1}$ isomorphic to $1+p\mathbb{Z}_{p}$. An example of such a module is the inverse limit of the $p$-part of class groups of cyclotomic fields.

The Main Conjecture of Iwasawa Theory

Let $A_{n}$ be the Sylow p-subgroup of the ideal class group of $\mathbb{Q}(\mu_{p^{n}})$. Let $X_{\infty}$ be the inverse limit of the $A_{n}$ under the norm maps. Then $X_{\infty}$ is a module over the Iwasawa algebra $\mathbb{Z}_{p}[[\Gamma]]$.

There is also an action of $\mathbb{Z}_{p}[[\Delta]]$ on $X_{\infty}$, where $\Delta=\mathrm{Gal}(\mathbb{Q}(\mu_{p})/\mathbb{Q})$. Let $\delta$ be a generator of $\Delta$ and let $a$ be such that $\delta(\zeta_{p})=\zeta_{p}^{a}$. Then $X_{\infty}$ breaks up into eigenspaces

$$X_{\infty}^{(i)}=\lbrace x\in X_{\infty}\vert\delta(x)=a^{i}x\rbrace$$

The main conjecture of Iwasawa theory then says that for odd $i$, the characteristic ideal of $X_{\infty}^{(i)}$ is generated by a power series $g_{i}$ such that

$$g_{i}(v^{s}-1)=L_{p}(\omega^{1-i},s)$$

where $v$ is the topological generator of $1+p\mathbb{Z}_{p}$ which is the evaluation of the p-adic cyclotomic character on the topological generator of $\Gamma$ and $L_{p}(\omega^{1-i},s)$ is the Kubota-Leopoldt p-adic zeta function.

The main conjecture of Iwasawa theory was proved by Mazur & Wiles 1984. It generalizes the Herbrand-Ribet theorem. The method of proof for the main conjecture of Iwasawa theory also follows similar ideas to the proof of the converse to Herbrand’s theorem in Ribet76.

Relation to Arithmetic Topology

Via the 3-manifold/number field analogy of arithmetic topology, Iwasawa theory can be seen as the analog of Alexander-Fox theory (see sec. 7 of Morishita).

References

• Romyar Sharifi, Modular Curves and Cyclotomic Fields pdf

• Ralph Greenberg (2001), Iwasawa theory—past and present, in Miyake, Katsuya, Class field theory—its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math. 30, Tokyo: Math. Soc. Japan, pp. 335–385 (ps file).

• Ralph Greenberg, Topics in Iwasawa Theory, (online book in process of being written).

• Kazuya Kato, Iwasawa theory and generalizations, (ICM 2006 talk).

• Masanori Morishita, Analogies between Knots and Primes, 3-Manifolds and Number Rings, (arxiv)

See also:

• Wikipedia, Iwasawa theory

The main conjecture of Iwasawa theory was proved in

• Barry Mazur and Andrew Wiles, Class fields of abelian extensions of $\mathbb{Q}$, Invent. Math. 76 (1984) 179–330 [eudml:143124, MR0742853]

The proof of the converse to Herbrand’s theorem is in

• Ken Ribet, A modular construction of unramified p-extensions of $\mathbb{Q}(\mu_{p})$, Invent. Math. 34 (1976), 151–162. MR0419403