In Bernoulli Numbers, Fermat’s Last Theorem, and the Riemann Zeta Function, we introduced the Kubota-Leopold -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 -adic modular forms.
Let us start with a little introduction to Iwasawa theory. Consider the Galois group , where is the extension of the rational numbers obtained by adjoining all the -th-power roots of unity to . This Galois group is isomorphic to , the group of units of the -adic integers .
The group decomposes into the product of a group isomorphic to and a group isomorphic to -th roots of unity. Let be the subgroup of this Galois group isomorphic to . The Iwasawa algebra is defined to be the group ring , which also happens to be isomorphic to the power series ring .
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 -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 and gives an element of . This should remind us of measures and integrals in real analysis, except instead of our functions being on , they are on the group , and instead of taking values in , they take values in . 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 , and note that is a subset of . Suppose we have an element of the Iwasawa algebra. We define the corresponding measure by saying what it does to a function on . Note that if we extend this function linearly, we can evaluate it on the element of the Iwasawa algebra and get an element of . 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 , 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 -adic modular forms, in particular -adic Eisenstein series.
The reason modular forms are brought into this is that the value of the zeta function at shows up in the constant term in the Fourier expansion of the Eisenstein series :
where , as is common convention in the theory (hence the Fourier expansion is also known as the -expansion). This Eisenstein series is a modular form of weight . A similar relationship holds between the Kubota-Leopoldt -adic L-function and -adic Eisenstein series, the latter of which is an example of a -adic modular form. We will define this now. Let be a modular form defined over . This means that, when we consider its Fourier expansion
the coefficients are rational numbers. We define a -adic valuation on the space of modular form by taking the biggest power of among the coefficients , i.e.
We recall that the bigger the power of dividing a rational number, the smaller its -adic valuation. This lets us consider the limit of a sequence. A -adic modular form is the limit of a sequence of classical modular forms.
The weight of a -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 and , if the -adic valuation
for some , then the weights of and will be congruent mod .
This implies that the weight of a -adic modular form takes values in the inverse limit of , which is isomorphic to the product of and . Here is where measures come in – this space of weights can be identified with characters of , i.e. a weight is a function on 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 , we can build a -adic Eisenstein series of weight (recall that this is a limit of classical Eisenstein series):
We think of this as a “measure” that takes a weight (again recall that the weight is a character, i.e. a function on ) and gives a weight Eisenstein series, i.e an “Eisenstein measure“. But the value of the Kubota-Leopoldt zeta function at 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 (the weight ) and giving us an element of . 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. -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!
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