In number theory, we often study arithmetic objects over a single number field. For example, given a number field $K$, we might ask: what is the class group of $K$? Or given an elliptic curve $E/K$, we might ask: what is the rank of the Mordell-Weil group of $E$ over $K$? These questions are often very deep and hard to answer. Iwasawa theory is based on the counterintuitive insight that even though answering these questions over a single number field is hard, answering them over a *infinite tower* of number fields is often easier.

$\{ \text{arithmetic objects over a number field} \}\, \xleftrightarrow \, \{\text{arithmetic objects over an infinite tower of number fields} \}$

For example, instead of asking: "what is the class group of this number field?", Iwasawa theory would ask, given an infinite tower of number fields

$K_1 \subset K_2 \subset K_3 \subset \dots$

how does the class group of $K_n$ *grow* as $n$ goes to infinity? Or given an elliptic curve $E/K_1$, how does the rank of the group $E(K_n)$ *grow* as $n$ goes to infinity? The fundamental insight of Iwasawa theory is that these growth questions are often *easier* to answer than their counterparts over single number fields.

The first proof of concept of this philosophy was given by Iwasawa in the following now famous theorem. For an integer $m \geq 1$, let $\mathbb{Q}(\zeta_m)$ denote the $m$-th cyclotomic field.

**Theorem.** (Iwasawa) Let $p$ be a prime. Consider the tower of cyclotomic fields

$\mathbb{Q}(\zeta_p) \subset \mathbb{Q}(\zeta_{p^2}) \subset \mathbb{Q}(\zeta_{p^3}) \subset \mathbb{Q}(\zeta_{p^3}) \subset \dots$

Let $p^{e_n}$ be the exact power of $p$ dividing the class number of $\mathbb{Q}(\zeta_{p^n})$. Then there exist integers $\mu, \lambda, \nu \geq 0$ such that

$e_n = \mu p^n + \lambda n + \nu$

for all $n \geq 0$.

Why is this theorem so interesting? Well, *nobody knows how to compute class groups*. The class groups of cyclotomic fields, especially, are basically impossible to calculate by naive methods once the fields get large. So the fact that you can say *anything* about them is quite amazing. Most people could not calculate the class group of even a single cyclotomic field. Iwasawa arranged them in an infinite tower and calculated their class number in one shot.

Before we proceed, it's worth pointing out what Iwasawa's theorem *doesn't* tell us. First, the theorem doesn't tell us anything about the group structure of the class groups. It only tells us about the class numbers. Furthermore, it also doesn't tell us the size of the *entire* class group, only the power of $p$ dividing the class number. Lastly, it doesn't tell us what the integers $\mu, \lambda, \nu$ are explicitly; it is a purely abstract result. Even given these caveats, however, Iwasawa's theorem is pretty amazing.

Consider $p=5$, so we have the number fields

$\mathbb{Q}(\zeta_5) \subset \mathbb{Q}(\zeta_{5^2}) \subset \mathbb{Q}(\zeta_{5^3}) \subset \mathbb{Q}(\zeta_{5^4}) \subset \dots$

It turns out that $\mu=\lambda=\nu = 0$ in Iwasawa's formula. So if $5^{e_n}$ is the power of $5$ dividing the class number of $\mathbb{Q}(\zeta_{5^n})$, we have $e_n=0$ for all $n$. In other words, $5$ is coprime to the class number of $\mathbb{Q}(\zeta_{5^n})$ for all $n$.

Now consider $p=37$, so we have the number fields

$\mathbb{Q}(\zeta_{37}) \subset \mathbb{Q}(\zeta_{37^2}) \subset \mathbb{Q}(\zeta_{37^3}) \subset \mathbb{Q}(\zeta_{37^4}) \subset \dots$

It turns out that $\mu=0$ and $\lambda=\nu =1$ in Iwasawa's formula. So if $37^{e_n}$ is the power of $37$ dividing the class number of $\mathbb{Q}(\zeta_{37^n})$, then

$e_n = n+1$

for all $n\geq 0$. That is, the power of $37$ dividing the class number of $\mathbb{Q}(\zeta_{37^n})$ grows linearly.

The proof of Iwasawa's theorem is incredibly beautiful and it sets up a general strategy for proving growth theorems in infinite towers. In this post, I'll sketch the strategy of the proof, without giving all the details. The full proof is given in Chapter 13 of Lawrence Washington's book "Introduction to Cyclotomic Fields". I highly recommend that you read this chapter because it's beautifully writen and it was what first made me fall in love with Iwasawa theory.

So the setup is that we have a tower of number fields

$\mathbb{Q}(\zeta_p) \subset \mathbb{Q}(\zeta_{p^2}) \subset \mathbb{Q}(\zeta_{p^3}) \subset \mathbb{Q}(\zeta_{p^3}) \subset \dots$

Let $X_n$ denote the $p$-Sylow subgroup of the class group $\text{Cl}(\mathbb{Q}(\zeta_{p^n}))$. There is a norm map $N: \mathbb{Q}(\zeta_{p^{n+1}}) \to \mathbb{Q}(\zeta_{p^n})$ and this induces a norm map on the class groups $X_{n+1} \to X_n$ for all $n$. So we get the following picture:

$\dots \to X_{n+3} \to X_{n+2} \to X_{n+1} \to X_{n} \to \dots$

where the arrows are the norm maps. Put

$X_{\infty} = \varprojlim X_n,$

where the inverse limit is taken with respect to the norm maps. An element of $X_{\infty}$ is an infinite sequence $(x_1, x_2, x_3, x_4, \dots)$ where $x_n \in X_n$ and the norm of $x_n$ is equal to $x_{n+1}$.

Iwasawa's insight is that even though each $X_n$ is very mysterious, the a priori more complex object $X_{\infty}$ is actually *easier* to study. In fact, we can study $X_{\infty}$ using tools from commutative algebra. To do this, first note that $X_n$ is a $\Z_p$ module because it is $p$-primary abelian group. Furthermore, $X_n$ has an action of the Galois group $\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})$. Therefore, $X_n$ is a module over the *group ring* $\Z_p [\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})]$.

Taking inverse limits, this means that $X_{\infty} = \varprojlim X_n$ is a module over the inverse limit of group rings: $\varprojlim \Z_p [\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})]$. The first inverse limit is taken with respect to the norm maps on class groups. The second inverse limit is taken with respect to the quotient maps on the Galois groups.

So $X_{\infty}$ is module over the mysterious ring $\varprojlim \Z_p [\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})]$. We will then study the $X_{\infty}$ using general commutative algebra results about modules over the ring $\varprojlim \Z_p [\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})]$. To do this, we will use the following important fact.

**Fact.** There is an isomorphism of $\Z_p$-modules

$\varprojlim_n \, \Z_p [\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})] \simeq \Z_p \llbracket T \rrbracket,$

where $\Z_p \llbracket T \rrbracket$ is the ring of formal power series in $T$ with coefficients in $\Z_p$.

From now on, *the only fact we will remember is this:* $X_{\infty}$ is a $\Z_p \llbracket T \rrbracket$-module. Our main input will be the below structure theorem from commutative algebra.

Put $\Lambda = \Z_p \llbracket T \rrbracket$. We'll now give a structure theorem for finitely-generated $\Lambda$-modules that is reminiscent of the structure theorem of finitely generated modules over a PID. There are two things that we need to state the theorem:

- A polynomial $f \in \Lambda$ is
*distinguished*if when you reduce it mod $p$, only the highest degree term remains. - If $M_1$ and $M_2$ are $\Lambda$-modules, then a
*pseudo-isomorphism*is a map $M_1 \to M_2$ with finite kernel and cokernel. If there is such a pseudo-isomorphism, then we write $M_1 \sim M_2$,

* Structure Theorem for $\Lambda$-modules.* (Iwasawa-Serre) Let $M$ be a finitely-generated torsion $\Lambda$-module. Then there is a pseudo-isomorphism

$M \sim \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}}\,\, \oplus \,\, \bigoplus_{j=1}^m \dfrac{\Lambda}{f_j},$

where the $f_j$ are distinguished polynomials.

One can show that $X_{\infty}$ is indeed finitely-generated and torsion, so the structure theorem gives us a pseudo-isomorphism:

$X_{\infty} \sim \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}}\,\, \oplus \,\, \bigoplus_{j=1}^m \dfrac{\Lambda}{f_j},$

where the $f_j$ are distinguished polynomials.

We now have an abstract result about $X_{\infty}$. How do we extract information about the structure of $X_n$? Our key will be the following fact:

* Proposition.* There is an isomorphism of $\Lambda$-modules:

$X_{\infty}/((1+T)^{p^n}-1) \simeq X_n.$

This proposition is hugely important because it allows us to recover $X_n$ from $X_{\infty}$. It tells us that to calculate $|X_n|$, it is enough to calculate $|X_{\infty}/((1+T)^{p^n}-1)|$ for all $n$. We will do this using the structure theorem. Given that we have a pseudo-isomorphism:

$X_{\infty} \sim \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}}\,\, \oplus \,\, \bigoplus_{j=1}^m \dfrac{\Lambda}{f_j},$

let's calculate the right-hand side of the above equation modulo $(1+T)^{p^n}-1$.

- First, we have

$\left|\left( \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}} \right )/\,((1+T)^{p^n}-1) \right| = p^{\left(\sum_{i=1}^n e_i\right)\,p^n}.$

- Next, we have

$\left|\left( \bigoplus_{i=1}^n \dfrac{\Lambda}{f_j} \right )/\,((1+T)^{p^n}-1) \right| = p^{\left(\sum_{j=1}^m \text{deg } f_j\right)\,n}.$

One can show that the "pseudo-isomorphism" gives us a factor of $p^\nu$ for some constant $\nu$. So in total, we get that

$|X_n| = |X_{\infty}/((1+T)^{p^n}-1)|=p^{\left(\sum_{i=1}^n e_i\right)\,p^n + \left(\sum_{j=1}^m \text{deg } f_j \right) \, n + \nu}$.

Putting $\mu = \sum_{i=1}^n e_i$ and $\lambda = \left(\sum_{j=1}^m \text{deg } f_j \right)$, we get

$|X_n| = p^{\mu p^n + \lambda n + \nu}$

for all $n \geq 0$. This proves Iwasawa's theorem.