The Iwasawa Main Conjecture

The Iwasawa main conjecture is a deep connection between two objects: class groups of cyclotomic fields and special values of the Reimann zeta function. The former object is algebraic: it is a class group of a number field. The latter is analytic: it is the value of a complex function at certain special points. The main conjecture, therefore (whatever it is), spells out an interesting connection between an algebraic and an analytic object.

Kummer's Theorem

The first indication that such a connection might exist is a theorem of Kummer.

Theorem. (Kummer) Let pp be a prime number. Then pp divides the class number of Q(ζp)\mathbf{Q}(\zeta_p) if and only if pp divides the numerator of ζ(r)\zeta(r) for some integer r=1,3,5,r=-1,-3,-5, \dots.

As an example, set p=691p=691. Then 691691 divides the numerator of ζ(11)=69132760\zeta(-11)=\frac{691}{32760}. So Kummer's theorem tells us that 691691 divides the class number of Q(ζ691)\mathbb{Q}(\zeta_{691}). This is already quite interesting because calculating class numbers by hand is basically impossible. This theorem says that the Reimann zeta function can be used to obtain this information in a different way.

But what is the meaning of 11-11 in ζ(11)\zeta(-11)? It turns out that 11-11 encodes the Galois action on the class group of Q(ζ691)\mathbb{Q}(\zeta_{691}). Precisely, the class group Cl (Q(ζ691))\text{Cl }(\mathbb{Q}(\zeta_{691})) has a natural action of the Galois group Gal (Q(ζ691)/Q) \text{Gal }(\mathbb{Q}(\zeta_{691})/\mathbb{Q}). And this Galois group acts on Cl Q(ζ691)\text{Cl }\mathbb{Q}(\zeta_{691}) via χ11\chi^{-11}, where χ\chi is the cyclotomic character. In other words, we have an inclusion of Galois modules

Cl (Q(ζ691))(Z/691Z)(11),\text{Cl }(\mathbb{Q}(\zeta_{691})) \supset (\mathbb{Z}/691\mathbb{Z})(-11),

where the (11)(-11) means that Galois acts via χ11\chi^{-11}.

Herbrand-Ribet Theorem

This example actually holds in general, and this was proved by Herbrand and Ribet.

Theorem. (Herbrand-Ribet) Let pp be a prime number and let r=1,3,5,r=-1,-3,-5, \dots. Then

Cl (Q(ζp))(Z/pZ)(r)    pζ(r).\text{Cl }(\mathbb{Q}(\zeta_p)) \supset (\mathbb{Z}/p\mathbb{Z})(-r)\iff p \,\vert \, \zeta(r).

The slogan here is:

zeta value= class group with Galois action.\text{zeta value}=\text{ class group with Galois action}.

It actually turns out that much more is true. To see that, I'll rephrase the Herbrand-Ribet Theorem above as follows. Let Cl(r)(Q(ζp))\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p)) be the χr\chi^r-eigenspace of Cl(Q(ζp))\text{Cl}(\mathbb{Q}(\zeta_p)) (i.e: it is the subgroup of Cl(Q(ζp))\text{Cl}(\mathbb{Q}(\zeta_p)) where Galois acts via χr\chi^r). Then Herbrand-Ribet equivalently says:

Theorem. Let pp be a prime number and let r=1,3,5,r=-1,-3,-5, \dots. Then

ordp(#Cl(r)(Q(ζp)))>0    ordp(ζ(r))>0\text{ord}_p(\#\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p))) > 0 \iff \text{ord}_p(\zeta(-r))>0.

In truth, however, it turns out that the ordp\text{ord}_p of both sides are in fact equal. That is, one can show that

ordp(#Cl(r)(Q(ζp)))=ordp(ζ(r))\text{ord}_p(\#\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p))) = \text{ord}_p(\zeta(-r)).

This however is much harder to prove, and requires the full strength of the Iwasawa main conjecture. In that sense, the Iwasawa main conjecture (whatever it is) is a strengthening of the Herbrand-Ribet Theorem.

The Iwasawa Main Conjecture

To state the Iwasawa main conjecture, we have to do something which is a signature of Iwasawa theory: we have to state the formulate Herbrand-Ribet theorem in infinite towers. On the algebraic side, instead of looking at just one class group, look at class groups in an infinite tower of cyclotomic fields. On the analytic side, instead of looking at just one zeta value, we will look at infinitely many zeta values are pp-adically interpolate them into a pp-adic LL-function.

The Algebraic Side

For Herbrand-Ribet, we considered the group Cl(r)(Q(ζp))\text{Cl}^{(r)}(\mathbb{Q}(\zeta_p)), which is the χr\chi^r eigenspace of the class group of Q(ζp)\mathbb{Q}(\zeta_p). Now we will do everything in Zp\mathbb{Z}_p-extensions.

Consider the Zp\mathbb{Z}_p-extension Q(ζp/Q(ζp)\mathbb{Q}(\zeta_{p^{\infty}}/\mathbb{Q}(\zeta_{p}) with layers given by:

Q(ζp)Q(ζp2)Q(ζp3)Q(ζp)\mathbb{Q}(\zeta_p) \subset \mathbb{Q}(\zeta_{p^2}) \subset \mathbb{Q}(\zeta_{p^3}) \subset \dots \subset \mathbb{Q}(\zeta_{p^{\infty}}).

Then for every n0n \geq 0, consider we can consider the χr\chi^r-eigenspace Cl(r)(Q(ζpn))\text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n})). The central object of study will be the inverse limit of these groups:

X(r)=limnCl(r)(Q(ζpn)).X_{\infty}^{(r)} = \varprojlim_n \text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n})).

Then X(r)X_{\infty}^{(r)} is a module over Λ=ZpGal (Q(ζp/Q(ζp))\Lambda = \mathbb{Z}_p \llbracket \text{Gal }(\mathbb{Q}(\zeta_{p^{\infty}}/\mathbb{Q}(\zeta_{p})) \rrbracket. And X(r)X_{\infty}^{(r)} is also finitely generated and torsion as a Λ\Lambda-module. There is a non-canonical isomorphism ΛZpT\Lambda \simeq \mathbb{Z}_p \llbracket T\rrbracket. So by the structure theorem for finitely generated torsion Λ\Lambda-modules, we have a pseudoisomorphism:

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

where the fjZpTf_j \in \mathbb{Z}_p \llbracket T\rrbracket are distinguished polynomials1.

Two quantities of interest:

  1. Define the μ\mu-invariant to be μ=e1++en\mu = e_1 + \dots + e_n.
  2. Define the λ\lambda-invariant to be λ=deg f1++deg fm\lambda = \text{deg }f_1 + \dots + \text{deg }f_m.

Then we have Iwasawa's growth formula: there is an integer ν0\nu\geq 0 such that

Cl(r)(Q(ζpn))=pμpn+λn+ν,|\text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n}))| = p^{\mu p^n + \lambda n + \nu },

for all n0n \geq 0. So the quantities μ\mu and λ\lambda in the structure theorem have a concrete meaning in terms of class numbers of the intermediate layers of the tower.

We can wrap up μ\mu and λ\lambda in a quantity called the characteristic ideal. The characteristic ideal of X(r)X_{\infty}^{(r)} is defined as:

char X(r)=(pe1++enf1fm)ZpT\text{char }X_{\infty}^{(r)} = (p^{e_1 + \dots + e_n} f_1 \dots f_m) \subset \Z_p \llbracket T \rrbracket.

That is, you take all the "denominators" in the structure theorem, multiply them together. The ideal generated by that element in ZpT\Z_p \llbracket T \rrbracket is the characteristic ideal. Note that we can recover μ\mu and λ\lambda from the characterstic ideal.

The Analytic Side

On the algebraic side, we looked at infinitely many number fields and "strung together" their class groups by taking an inverse limit. On the analytic side we will look at infinitely many zeta values and we will string them together by pp-adically interpolating them. Doing this will create a pp-adic LL-function, which is the main object of the analytic side of the main conjecture.

We discussed pp-adic LL-functions in detail in a previous post. I'd definitely recommend reading that post first before coming here. For here, I'll just recall the main theorem from that post, which proves the existence and uniqueness of the pp-adic Riemann zeta function.

Theorem. (Kubota-Leopoldt) Let pp be a prime, and let r=1,3,5,r = -1, -3, -5, \dots be a negative odd integer. There exists a unique pp-adically continuous function ζp(r):ZpZp\zeta_p^{(r)}: \Z_p \to \Z_p with the following interpolation property:

Define the set Sr={nZ<0:nrmod(p1)}S_r = \{ n \in \Z_{<0}: n \equiv r \mod (p-1) \}. Then for all sSrs \in S_r, we have

ζp(s)=(1ps)ζ(s)\zeta_p(s) = (1-p^{-s}) \zeta(s).

This function is called the (rr-th branch of the) Kubota-Leopoldt pp-adic zeta function. The set SrS_r consists of the "points of interpolation" of ζp(r)\zeta_p^{(r)}. At these points, the function ζp(r)\zeta_p^{(r)} coincides with the Riemann zeta function with the Euler factor at pp removed.

If any of this feels unfamiliar, check out this post where I explained this theorem much more slowly.

The Iwasawa main conjecture relates the module X(r)X_{\infty}^{(r)} to the pp-adic LL-function ζp(r)\zeta_p^{(r)}:

X(r)Iwasawa Main Conjectureζp(r)X_{\infty}^{(r)} \xleftrightarrow{\text{Iwasawa Main Conjecture}}\, \zeta_p^{(r)}.

The problem is that these two objects live in different worlds: the module X(r)X_{\infty}^{(r)} is an object from commutative algebra whereas ζp(r)\zeta_p^{(r)} is a pp-adic analytic function. So it's not even clear how to state a precise conjecture relating them. It was Iwasawa who realized that ζp(r)\zeta_p^{(r)} can be expressed as a power series in ZpT\mathbb{Z}_p \llbracket T\rrbracket, which will allow us to view it as an object in commutative algebra.

Theorem. (Iwasawa) Let pp be a prime and let r=1,3,5,r=-1, -3, -5, \dots be an odd negative integer. Then there is a unique power series ζ(T)ZpT\zeta(T) \in \mathbb{Z}_p \llbracket T\rrbracket satisfying

Gr((1+p)s1)=ζp(r)(s)G_r((1+p)^s-1) = \zeta_p^{(r)}(s).

for all sZps \in \Z_p.

By abuse of notation, we also call Gr(T)G_r(T) the (rthr^{th} branch of) the Kubota Leopoldt pp-adic zeta function. We will no longer think of the pp-adic zeta function as a literal function ZpZp\Z_p \to \Z_p, but we will view it as a power series in ZpT\Z_p \llbracket T \rrbracket.

The Main Conjecture

On the algebraic side, we have the characteristic ideal char (X(r))\text{char }(X_{\infty}^{(r)}). On the analytic side, we have the pp-adic zeta function Gi(T)G_i(T). We can look at the ideal (Gi(T))ZpT(G_i(T)) \subset \mathbb{Z}_p \llbracket T\rrbracket generated by this pp-adic zeta function. The main conjecture asserts that these two ideals are equal.

Iwasawa Main Conjecture. Let pp be a prime and let r=1,3,5,r = -1, -3, -5, \dots be a negative odd integer. Then:

char (X(r))=(Gr(T))\text{char }(X_{\infty}^{(r)}) = (G_r(T)).

When I first saw this, I understood basically nothing. So I want to take it apart a bit to understand it better. First, I found it helpful to phrase this in terms of power series instead of ideals. We had a pseudo-isomorphism:

X(r)i=1nΛpeij=1mΛfj.X_{\infty}^{(r)} \sim \bigoplus_{i=1}^n \dfrac{\Lambda}{p^{e_i}} \, \oplus \bigoplus_{j=1}^m \dfrac{\Lambda}{f_j}.

Also, by the Weierstrauss preparation theorem, we can factor Gr(T)G_r(T) as

Gr(T)=pku(T)f(T),G_r(T) = p^k \, u(T) \, f(T),

where u(T)u(T) is a unit and f(T)f(T) is a distinguished polynomial.

Iwasawa Main Conjecture. (Power Series Formulation) We have

pe1++enf1fmGr(T)p^{e_1 + \dots + e_n} f_1 \dots f_m \, \sim \, G_r(T).

where \sim indicates equality upto a unit in ZpT\Z_p \llbracket T \rrbracket. In particular, e1++en=ke_1 + \dots + e_n = k and f1fmff_1 \dots f_m \sim f.

This might seem kind of opaque (I definitely thought so at first), but it has a very concrete consequence about μ\mu- and λ\lambda-invariants. To see it, let's remember Iwasawa's growth formula: there is an integer ν0\nu\geq 0 such that

Cl(r)(Q(ζpn))=pμpn+λn+ν,|\text{Cl}^{(r)}(\mathbb{Q}(\zeta_{p^n}))| = p^{\mu p^n + \lambda n + \nu },

for all n0n \geq 0. The main conjecture then implies that:

Consequence of IMC.

  1. The quantity μ\mu in the above formula is the power of pp dividing Gr(T)G_r(T).
  2. And the quantity λ\lambda in the above formula is the degree of the of pp dividing Gr(T)G_r(T).

So we can read off μ\mu and λ\lambda right from the pp-adic zeta-function. This is really weird! The pp-adic zeta-function interpolates special values of the Riemann zeta function. Why on earth should we be able to read off μ\mu and λ\lambda-invariants for class groups from these zeta functions?

The Main Conjecture is now a theorem!

Regardless of how mysterious the statement is, Mazur and Wiles were actually able to prove the main conjecture in a brilliant 1984 Inventiones paper.

Theorem. (Mazur-Wiles) The Iwasawa Main Conjecture is true.

Footnotes

  1. A polynomial fZpTf \in \mathbb{Z}_p \llbracket T\rrbracket is distinguished if when you reduce it mod pp, only the highest degree term remains.