Why did Iwasawa work so hard?

This post is very rough and is only to remind me of my chain of thought. Read at your own peril.

The first example of Iwasawa theory most people see is the historically first one of class groups for the cyclotomic $latex\mathbb Z_p$ tower of $\mathbb Q$. The reason the idea works at all is that somehow, passing to the limit reduces the “amount of bad stuff” that can happen and the key to this is a sort of control theorem where, if we denote the inverse limit of class groups by $X$ which has an action of $\Lambda \cong \mathbb Z_p[[t]]$, then Iwasawa shows that the class groups $X_n$ at finite levels can be recovered by essentially doing $X/v_n X$ where $v_n = (1+t)^{p^n} - 1$. This step very crucially uses class field theory in the identification of the class group with the maximal unramified extension.

Now one might object that perhaps we don’t need to work so hard. After all, $X = \lim X_n$ where the limit is over the algebras $\mathbb Z_p[[t]]/v_n \cong \mathbb Z_p[\theta]/(\theta^{p^n} - 1)$ where $\theta = 1+t$ and it seems natural that $X/v_n X = X_n$. Unfortunately (or fortunately), this is not true at all and is where Iwasawa theory gets it’s great power from. The purpose of this post is to record how bad things can go without any control theorems.

Let us denote by $\phi_n(x)$ the cyclotomic polynomial of order $p^n$ so that $v_n$ is essentially a product of these and while $\mathbb Z_p[[t]]/v_n(t)$ is not congruent to $\prod_{i=1}^n\mathbb Z_p[[t]]/\phi_i(1+t)$, it almost is in the sense that if we have an element (or module) over each $\mathbb Z_p[[t]]/\phi_i(1+t)$ which agree after reducing modulo the respective maximal ideals, then they can be glued back to an actual element (or module) over $Z_p[[t]]/v_n(t)$. This is some version of the chinese remainder theorem.

For modules, the condition about reducing modulo maximal ideals corresponds to saying that over the special point of each ring, the modules have the same $\mathbb F_p$ dimension. Since each ring is itself a DVR, or close to it, modules over it are classified by the order of their torsion so let me do an easy example where we only consider two of the rings, say corresponding to $\zeta_{p^k}, \zeta_{p^{k+1}}$ with torsion orders $1,1$ and rank 1.

Idea: We use the cyclotomic polynomials to kill all but one of the relations over the various rings and introduce appropriate torsion but this leads to problems with dividing by $p$ so we define new variables that act as though they result from dividing by $p$.

Construction: We can define a module over $\mathbb Z_p[[t]]/\phi_k(1+t)\phi_{k+1}(1+t)$ in the following way:

Let it be generated by two elements $e,f$ with the relation $pe = f$ and we impose the conditions that $\phi_k(1+t)e = 0$ and $\phi_l(1+t)e = 0$. The point is that sending $1+t \to \zeta_{p^k}$, the first condition is meaningless while $\phi{k+1}(\zeta_{p^k})=p$ and so the relations over $\mathbb Z_p[[t]]/\phi_{k}(t)$ are $pe = f, pe = 0$ or equivalently, it is generated by $e$ and $pe = 0$.

Similarly, sending $1+t \to \zeta_{p^{k+1}}$, the relations are of the form $\pi e = 0$ for $\pi$ a uniformizer of $\mathbb Z_p[zeta_p]$ or something like it and over $\mathbb F_p$, one can check that this is rank $1$.

What goes wrong in the Iwasawa tower: Using this construction, we can, at each step of the inverse limit, make whatever arbitrary order of torsion we want (keeping the mod p rank constant). However, in order to do this, we will have to introduce larger and larger products of cyclotomic polynomials and if we don’t stop at some stage, the relations will go to zero in the limit.

Therefore, we can have arbitrary torsion at each stage but at the cost of having no torsion whatsoever in the limit and then clearly, the control theorem won’t work – the quotients will continue to be free. This is why it’s crucial that Iwasawa could establish a “control theorem”.