In Bernoulli Numbers, Fermat’s Last Theorem, and the Riemann Zeta Function, we introduced the Kubota-Leopold $p$-adic L-function, which encodes the congruences discovered by Kummer between special values of the Riemann zeta function. In this post, we will connect them to Iwasawa theory and $p$-adic modular forms.

Let us start with a little introduction to Iwasawa theory. Consider the Galois group $\text{Gal}(\mathbb{Q}(\mu_{p^{\infty}})/\mathbb{Q})$, where $\mathbb{Q}(\mu_{p^{\infty}})$ is the extension of the rational numbers $\mathbb{Q}$ obtained by adjoining all the $p$-th-power roots of unity to $\mathbb{Q}$. This Galois group is isomorphic to $\mathbb{Z}_{p}^{\times}$, the group of units of the $p$-adic integers $\mathbb{Z}_p$.

The group $\mathbb{Z}_{p}^{\times}$ decomposes into the product of a group isomorphic to $1+p\mathbb{Z}_{p}$ and a group isomorphic to $(p-1)$-th roots of unity. Let $\Gamma$ be the subgroup of this Galois group isomorphic to $1+p\mathbb{Z}_{p}$. The Iwasawa algebra is defined to be the group ring $\mathbb{Z}_{p}[[\Gamma]]$, which also happens to be isomorphic to the power series ring $\mathbb{Z}_{p}[[T]]$.

The interest in the Iwasawa algebra comes from the fact that many important objects of interest in number theory are modules over the Iwasawa algebra, and such modules have a structure that makes them easy to study. For instance, the inverse limit of the p-part of the ideal class groups of cyclotomic fields is such a module. The “main conjecture of Iwasawa theory“, a high-powered version of Kummer’s theorem that relates ideal class groups and Bernoulli numbers, describes this module. Namely, the main conjecture of Iwasawa theory states that as a module over the Iwasawa algebra, the inverse limit of the p-part of the ideal class groups of cyclotomic fields has a characteristic ideal generated by none other than the Kubota-Leopoldt $p$-adic L-function!

Let us describe more the relation between the Iwasawa algebra and the Kubota-Leopoldt zeta function by relating them to measures. Our measure here takes functions on the group $\mathbb{Z}_p^{\times}$ and gives an element of $\mathbb{Z}_{p}$. This should remind us of measures and integrals in real analysis, except instead of our functions being on $\mathbb{R}$, they are on the group $\mathbb{Z}_{p}^{\times}$, and instead of taking values in $\mathbb{R}$, they take values in $\mathbb{Z}_{p}$. This is just an example of a more general kind of measure.

Now these measures are actually in one-to-one correspondence with the elements of the Iwasawa algebra!

The Iwasawa algebra is $\mathbb{Z}_{p}[[\Gamma]]$, and note that $\Gamma$ is a subset of $\mathbb{Z}_{p}^{\times}$. Suppose we have an element of the Iwasawa algebra. We define the corresponding measure by saying what it does to a function $f$ on $\mathbb{Z}_{p}^{\times}$. Note that if we extend this function linearly, we can evaluate it on the element of the Iwasawa algebra and get an element of $\mathbb{Z}_{p}^{\times}$. Thus we define our measure by evaluation. The other direction is a bit more involved, but given the measure, we build an element of the Iwasawa algebra by exploiting the profinite nature of $\mathbb{Z}_{p}^{\times}$, which means the measure was built from functions on the finite pieces of it.

Now we know how the Iwasawa algebra and measures are related, what about the Kubota-Leopoldt zeta function? For those we must now take a detour through $p$-adic modular forms, in particular $p$-adic Eisenstein series.

The reason modular forms are brought into this is that the value of the zeta function at $1-k$ shows up in the constant term in the Fourier expansion of the Eisenstein series $G_{k}$:

$\displaystyle G_{k}(\tau):=\frac{\zeta(1-k)}{2}+\sum_{n=1}^{\infty}\left(\sum_{d\vert n}d^{k-1}\right)q^{n}$

where $q=e^{2\pi i \tau}$, as is common convention in the theory (hence the Fourier expansion is also known as the $q$-expansion). This Eisenstein series $G_{k}$ is a modular form of weight $k$. A similar relationship holds between the Kubota-Leopoldt $p$-adic L-function and $p$-adic Eisenstein series, the latter of which is an example of a $p$-adic modular form. We will define this now. Let $f$ be a modular form defined over $\mathbb{Q}$. This means that, when we consider its Fourier expansion

$\displaystyle f(\tau)=\sum_{n=0}^{\infty}a_{n}q^{n}$,

the coefficients $a_{n}$ are rational numbers. We define a $p$-adic valuation on the space of modular form by taking the biggest power of $p$ among the coefficients $a_{n}$, i.e.

$\displaystyle v_{p}(f)=\inf_{n} v_{p}(a_{n})$

We recall that the bigger the power of $p$ dividing a rational number, the smaller its $p$-adic valuation. This lets us consider the limit of a sequence. A $p$-adic modular form is the limit of a sequence of classical modular forms.

The weight of a $p$-adic modular form is the limit of the weights of the classical ones of which it is the limit. Serre showed that for classical modular forms $f$ and $g$, if the $p$-adic valuation

$\displaystyle v(f-g)>=v(f)+m$

for some $m$, then the weights of $f$ and $g$ will be congruent mod $(p-1)p^m$.

This implies that the weight of a $p$-adic modular form takes values in the inverse limit of $\mathbb{Z}/(p-1)p^{m}\mathbb{Z}$, which is isomorphic to the product of $\mathbb{Z}_{p}$ and $(p-1)\mathbb{Z}$. Here is where measures come in – this space of weights can be identified with characters of $\mathbb{Z}_{p}^{\times}$, i.e. a weight $k$ is a function on $\mathbb{Z}_{p}^{\times}$and being such a function, it is an input for a measure!

Now, we will create a measure, with a bit of a twist. Given a weight $k$, we can build a $p$-adic Eisenstein series of weight $k$ (recall that this is a limit of classical Eisenstein series):

$\displaystyle G_{k}^{*}:=\varinjlim_{i}G_{k_{i}}$

We think of this as a “measure” that takes a weight $k$ (again recall that the weight $k$ is a character, i.e. a function on $\mathbb{Z}_{p}$) and gives a weight $k$ Eisenstein series, i.e an “Eisenstein measure“. But the value of the Kubota-Leopoldt zeta function at $1-k$ is the constant in the Fourier expansion! Therefore, if we take the constant term of this p-adic Eisenstein series, we have a good old measure, a recipe for taking a function on $\mathbb{Z}_{p}$ (the weight $k$) and giving us an element of $\mathbb{Z}_{p}$. But by our earlier discussion, this is an element of the Iwasawa algebra!

There are some subtleties I swept under the rug, but to summarize – important objects in number theory are modules over the Iwasawa algebra. $p$-adic L-functions which interpolate L-functions at special values are elements of the Iwasawa algebra.

This is a modern, high-powered version of Kummer’s discovery that relates certain ideal class groups and Bernoulli numbers (which are special values of the Riemann zeta function). The Eisenstein measure, which gives a p-adic modular form when evaluated at a certain weight, leads to the notion of a “Hida family“, a “p-adic family” of p-adic modular forms. But that discussion is for another time!

References:

Iwasawa theory on Wikipedia

Iwasawa algebra on Wikipedia