Iwasawa Theory

Introduction

The class group ${\mathrm{Cl}}_F$ of a number field $F$ is an object of central importance in number theory. It is a finite abelian group, and its order $h_F$ is known as the class number. In general, the explicit determination of $h_F$, let alone the structure of ${\mathrm{Cl}}_F$ as a finite abelian group, can be a difficult and computationally intensive task.

In the late 1950’s, Iwasawa initiated a study of the growth of class groups in certain towers of number fields. Given a tower $F=F_0\subset F_1\subset F_2\subset \cdots$ of Galois extensions of $F$, one asks if there is any regularity to the growth of $h_{F_n}$. The knowledge of this growth, in turn, can be used to say something about the structure of ${\mathrm{Cl}}_{F_n}$ as a finite abelian group. Iwasawa was concerned with towers such that $\mathrm{Gal}(F_{\infty }∕F)\cong {\mathbb{Z}}_p$ for some prime $p$, where $F_{\infty }={\bigcup }_n{F}_n$, known as ${\mathbb{Z}}_p$-extensions. He set $\mathrm{\Gamma }=\mathrm{Gal}(F_{\infty }∕F)$ and ${\mathrm{\Gamma }}_n=\mathrm{Gal}(F_n∕F)$, and let us suppose that $F_n$ is chosen to be (cyclic) of degree $p^n$ over $F$. For example, for odd $p$, the cyclotomic ${\mathbb{Z}}_p$-extension $F_{\infty }$ of $F$ is the largest subextension of $F({\mu }_{p^{\infty }})∕F$ with pro-$p$ Galois group.

The question of how $h_{F_n}$ grows in the tower defined by a ${\mathbb{Z}}_p$-extension is quite difficult, in particular as the order away from $p$ of ${\mathrm{Cl}}_{F_n}$ has little to do with the order away from $p$ of ${\mathrm{Cl}}_{F_{n+1}}$, other than the fact that the latter order is a multiple of the former. On the other hand, if we concentrate on the order $h_{F_n}^{(p)}$ of the Sylow $p$-sugroup $A_n$ of $F_n$, we have the following theorem of Iwasawa.

In the case that $F_{\infty }$is the cyclotomic ${\mathbb{Z}}_p$-extension, Iwasawa conjectured that the invariant $\mu$ in the theorem is $0$. Ferrero and Washington later proved this result for abelian extensions of $\mathbb{Q}$.

We have maps between the $p$-parts of class group in the tower in both directions $j_n:A_n\to A_{n+1}$, which takes the class of an ideal $𝔞$ to the class of the ideal it generates, and $N_n:A_{n+1}\to A_n$, which takes the class of an ideal to the class of its norm. Iwasawa considered the direct and inverse limits

under the $j_n$ and $N_n$, respectively. As each $A_n$ has the structure of a finite ${\mathbb{Z}}_p[{\mathrm{\Gamma }}_n]$-module through the standard action of ${\mathrm{\Gamma }}_n$ on ideal classes, both $X_{\infty }$and the Pontryagin dual $A_{\infty }^{\vee }={\mathrm{Hom}}_{\mathrm{cts}}(A_{\infty },{\mathbb{Q}}_p∕{\mathbb{Z}}_p)$ of $A_{\infty }$are finitely generated torsion modules over the competed ${\mathbb{Z}}_p$-group ring of $\mathrm{\Gamma }$:

The ring $\mathrm{\Lambda }={\mathbb{Z}}_p⟦\mathrm{\Gamma }⟧$ is known as the Iwasawa algebra, and it has a very simple structure. In fact, a choice of a topological generator $\gamma$ of $\mathrm{\Gamma }$ gives rise to an isomorphism

The following result on the structure of $\mathrm{\Lambda }$-modules allowed Serre to rephrase the theorem of Iwasawa.

From Serre’s theorem, we are able to deduce several important invariants of a finitely generated $\mathrm{\Lambda }$-module $M$. For instance, in the notation of the theorem, let us set

These are known as the $\lambda$ and $\mu$-invariants of $M$. Serre showed that these invariants for $X_{\infty }$ and $A_{\infty }^{\vee }$agree with the $\lambda$ and $\mu$ of Iwasawa’s theorem. An even more interesting invariant of $M$ is its characteristic ideal, given by

which we shall consider in a specific case shortly.

It is worth remarking here that one usually thinks of $X_{\infty }$ as a Galois group. Recall that the Artin reciprocity map provides an isomorphism between $A_n$ and the Galois group of the Hilbert $p$-class field $L_n$ of $F_n$, which is to say the maximal unramified abelian $p$-extension of $F_n$. Setting $L_{\infty }={\bigcup }_n{L}_n$, we have a canonical isomorphism $X_{\infty }\cong \mathrm{Gal}(L_{\infty }∕F_{\infty })$. The resulting action on $\mathrm{\Gamma }$ on $\mathrm{Gal}(L_{\infty }∕F_{\infty })$ is a conjugation action, given by a lift of $\mathrm{\Gamma }$ to a subsgroup of $\mathrm{Gal}(L_{\infty }∕F)$.

Let us focus now on the specific case that $F=\mathbb{Q}({\mu }_p)$, and let us take $F_{\infty }$to be the cyclotomic ${\mathbb{Z}}_p$-extension of $F$ for an odd prime $p$. In this setting, Iwasawa proved that his $\mu =\mu (X_{\infty })$ is zero.

We define the Teichmüller character $\omega :\mathrm{\Delta }\to {\mathbb{Z}}_p^{\times }$ by setting $\omega (\delta )$ for $\delta \in \mathrm{\Delta }$ to be the unique $(p-1)$st root of unity in ${\mathbb{Z}}_p$ such that

for any primitive $p$th root of unity ${\zeta }_p$.

As with $\mathrm{\Gamma }$, the Galois group $\mathrm{\Delta }=\mathrm{Gal}(F∕\mathbb{Q})$ will act on $X_{\infty }$. For any $i$, we may consider the eigenspace $X_{\infty }^{(i)}$ of $X_{\infty }$ on which every $\delta \in \mathrm{\Delta }$ acts through multiplication by ${\omega }^i(\delta )$. We have the following theorem of Herbrand and Ribet.

The interesting fact is that Bernoulli numbers and their generalizations appear as values of $L$-functions. Kubota and Leopoldt showed how that the $L$-values of certain characters at negative integers can be interpolated, in essence, by a function of ${\mathbb{Z}}_p$, denoted $L_p(\chi ,s)$ and known as a $p$-adic $L$-function.

Let us fix the particular generator $\gamma$ of $\mathrm{\Gamma }$ such that $\gamma (\zeta )={\zeta }^{1+p}$ for every $p$-power root of unity $\zeta$, and in particular the isomorphism of $\mathrm{\Lambda }$ with ${\mathbb{Z}}_p⟦T⟧$. Iwasawa made the following conjecture on the characteristic ideal of an eigenspace of $X$, which was later proven by Mazur and Wiles.

In fact, Mazur and Wiles proved a generalization of this to abelian extensions $F$ of $\mathbb{Q}$, and Wiles proved a further generalization to abelian extensions of totally real fields. This line of proof was primarily geometric in nature, and came by studying the action of the absolute Galois group of $F$ on the cohomology groups of modular curves. Rubin gave a proof of a rather different nature of a main conjecture for abelian extensions of imaginary quadratic fields, following work of Kolyvagin and Thaine, using a Galois cohomological tool known as an Euler system.

Let us end this introduction by mentioning the two of the major directions in which Iwasawa theory has expanded over the years. As a first and obvious course of action, one can replace our limits of class groups with more general objects. Via class field theory, we note that the Pontryagin dual $X_{\infty }^{\vee }$ may be identified with the kernel of the map

where $G_{F_{\infty },S}$ denotes the Galois group of the maximal extension of $F_{\infty }$ unramified outside $S$ and $S$ in this case is the set of primes of $F_{\infty }$ lying over $p$, and where $I_v$ is the inertia group at $v\in S$ in the absolute Galois group of $F_{\infty }$. That is, we have realized $X_{\infty }^{\vee }$ as what is known as a Selmer group. This generalizes nicely.

By way of the most interesting example, let $E$ be an elliptic curve over $F$ with ordinary reduction at $p$, and let $E[p^{\infty }]$ denote its $p$-power torsion (over $\overline{\mathbb{Q}}$). The Selmer group of $E$ over $F_{\infty }$is exactly

where $S$ is now the set of primes of $F_{\infty }$over $p$ or any primes of bad reduction of $E$. In the case that $F=\mathbb{Q}$, there is a corresponding main conjecture for the structure of $\mathrm{Sel}{(E∕F_{\infty })}^{\vee }$ in terms of a $p$-adic $L$-function of $E$. Great progress has been made on this particular main conjecture, due to successively more recent work of Rubin (for CM curves), Kato, and Skinner and Urban.

In the second generalization, one allows the Galois group $\mathrm{\Gamma }$ of the tower to take a more general form than ${\mathbb{Z}}_p$. The case that $\mathrm{\Gamma }$ is a $p$-adic Lie group, which is to say isomorphic to an open subgroup of ${\mathrm{GL}}_m({\mathbb{Z}}_p)$ for some $m\ge 1$, has come under the greatest consideration. In this case, main conjectures become more difficult to formulate, as the structure theory of $\mathrm{\Lambda }={\mathbb{Z}}_p⟦\mathrm{\Gamma }⟧$-modules is no longer simple. Still, in the past decade, such main conjectures have been formulated using $K$-theory as one of several tools. In the classical setting of limits of class groups, the corresponding main conjecture has been proven by Kakde and Ritter-Weiss.1