Iwasawa theory, p-adic L-functions, and p-adic modular forms

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

p-adic L-function on Wikipedia

Main conjecture of Iwasawa theory on Wikipedia

An introduction to Eisenstein measures by E. E. Eischen

Modular curves and cyclotomic fields by Romyar Sharifi

Desde Fermat, Lamé y Kummer hasta Iwasawa: Una introducción a la teoría de Iwasawa (in Spanish) by Álvaro Lozano-Robledo

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