What is Iwasawa theory?

In number theory, we often study arithmetic objects over a single number field. For example, given a number field KK, we might ask: what is the class group of KK? Or given an elliptic curve E/KE/K, we might ask: what is the rank of the Mordell-Weil group of EE over KK? 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.

{arithmetic objects over a number field}{arithmetic objects over an infinite tower of number fields}\{ \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

K1K2K3K_1 \subset K_2 \subset K_3 \subset \dots

how does the class group of KnK_n grow as nn goes to infinity? Or given an elliptic curve E/K1E/K_1, how does the rank of the group E(Kn)E(K_n) grow as nn 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 m1m \geq 1, let Q(ζm)\mathbb{Q}(\zeta_m) denote the mm-th cyclotomic field.

Theorem. (Iwasawa) Let pp be a prime. Consider the tower of cyclotomic fields

Q(ζp)Q(ζp2)Q(ζp3)Q(ζp3)\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 penp^{e_n} be the exact power of pp dividing the class number of Q(ζpn)\mathbb{Q}(\zeta_{p^n}). Then there exist integers μ,λ,ν0\mu, \lambda, \nu \geq 0 such that

en=μpn+λn+ν e_n = \mu p^n + \lambda n + \nu

for all n0n \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 pp dividing the class number. Lastly, it doesn't tell us what the integers μ,λ,ν\mu, \lambda, \nu are explicitly; it is a purely abstract result. Even given these caveats, however, Iwasawa's theorem is pretty amazing.

Examples

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

Q(ζ5)Q(ζ52)Q(ζ53)Q(ζ54)\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 μ=λ=ν=0\mu=\lambda=\nu = 0 in Iwasawa's formula. So if 5en5^{e_n} is the power of 55 dividing the class number of Q(ζ5n)\mathbb{Q}(\zeta_{5^n}), we have en=0e_n=0 for all nn. In other words, 55 is coprime to the class number of Q(ζ5n)\mathbb{Q}(\zeta_{5^n}) for all nn.

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

Q(ζ37)Q(ζ372)Q(ζ373)Q(ζ374)\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 μ=0\mu=0 and λ=ν=1\lambda=\nu =1 in Iwasawa's formula. So if 37en37^{e_n} is the power of 3737 dividing the class number of Q(ζ37n)\mathbb{Q}(\zeta_{37^n}), then

en=n+1e_n = n+1

for all n0n\geq 0. That is, the power of 3737 dividing the class number of Q(ζ37n)\mathbb{Q}(\zeta_{37^n}) grows linearly.

Proof of Iwasawa's Theorem

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.

The Setup

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

Q(ζp)Q(ζp2)Q(ζp3)Q(ζp3)\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 XnX_n denote the pp-Sylow subgroup of the class group Cl(Q(ζpn))\text{Cl}(\mathbb{Q}(\zeta_{p^n})). There is a norm map N:Q(ζpn+1)Q(ζpn)N: \mathbb{Q}(\zeta_{p^{n+1}}) \to \mathbb{Q}(\zeta_{p^n}) and this induces a norm map on the class groups Xn+1XnX_{n+1} \to X_n for all nn. So we get the following picture:

Xn+3Xn+2Xn+1Xn \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=limXn,X_{\infty} = \varprojlim X_n,

where the inverse limit is taken with respect to the norm maps. An element of XX_{\infty} is an infinite sequence (x1,x2,x3,x4,)(x_1, x_2, x_3, x_4, \dots) where xnXnx_n \in X_n and the norm of xnx_n is equal to xn+1x_{n+1}.

Commutative Algebra

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

Taking inverse limits, this means that X=limXnX_{\infty} = \varprojlim X_n is a module over the inverse limit of group rings: limZp[Gal(Q(ζpn)/Q)]\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 XX_{\infty} is module over the mysterious ring limZp[Gal(Q(ζpn)/Q)]\varprojlim \Z_p [\text{Gal}(\mathbb{Q}(\zeta_{p^n})/\mathbb{Q})]. We will then study the XX_{\infty} using general commutative algebra results about modules over the ring limZp[Gal(Q(ζpn)/Q)]\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 Zp\Z_p-modules

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

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

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

The Structure Theorem

Put Λ=ZpT\Lambda = \Z_p \llbracket T \rrbracket. We'll now give a structure theorem for finitely-generated Λ\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:

  1. A polynomial fΛf \in \Lambda is distinguished if when you reduce it mod pp, only the highest degree term remains.
  2. If M1M_1 and M2M_2 are Λ\Lambda-modules, then a pseudo-isomorphism is a map M1M2M_1 \to M_2 with finite kernel and cokernel. If there is such a pseudo-isomorphism, then we write M1M2M_1 \sim M_2,

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

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

where the fjf_j are distinguished polynomials.

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

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

where the fjf_j are distinguished polynomials.

Going from XX_{\infty} to XnX_n

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

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

X/((1+T)pn1)Xn.X_{\infty}/((1+T)^{p^n}-1) \simeq X_n.

This proposition is hugely important because it allows us to recover XnX_n from XX_{\infty}. It tells us that to calculate Xn|X_n|, it is enough to calculate X/((1+T)pn1)|X_{\infty}/((1+T)^{p^n}-1)| for all nn. We will do this using the structure theorem. Given that we have a pseudo-isomorphism:

Xi=1nΛpeij=1mΛfj,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)pn1(1+T)^{p^n}-1.

  1. First, we have

(i=1nΛpei)/((1+T)pn1)=p(i=1nei)pn.\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}.

  1. Next, we have

(i=1nΛfj)/((1+T)pn1)=p(j=1mdeg fj)n.\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 pνp^\nu for some constant ν\nu. So in total, we get that

Xn=X/((1+T)pn1)=p(i=1nei)pn+(j=1mdeg fj)n+ν|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 μ=i=1nei\mu = \sum_{i=1}^n e_i and λ=(j=1mdeg fj)\lambda = \left(\sum_{j=1}^m \text{deg } f_j \right), we get

Xn=pμpn+λn+ν|X_n| = p^{\mu p^n + \lambda n + \nu}

for all n0n \geq 0. This proves Iwasawa's theorem.