Describing the key Sub-algebra of the Bost-Connes System - Part III

Third in a three part series of me trying to describe the Bost-Connes algebra. Part I: $\mathbb{Q}$-Lattices and the Presentation. Part II: Equivalent description in terms of a Hecke Algebra. Part III: Describing the key Sub-algebra.

In my last post I briefly described how the Bost-Connes system was originally described as a Hecke algebra $\mathcal{H}_{\mathbb{C}}(P_{\mathbb{Q}}^{+},P_{\mathbb{Z}}^{+})$, where $P_R$ is the functor that takes an abelian ring $R$ to a group of 2 by 2 matrices over $R$. If you'll recall, we arrived at that Hecke algebra by taking the complexification of $\mathcal{H}_{\mathbb{Q}}(P_{\mathbb{Q}}^{+},P_{\mathbb{Z}}^{+})$. It turns out that this algebra is where all the KMS & Galois group intertwining-magic happens (which I hope to describe in my next post). The purpose in this post (and in section 4.4 of the monograph by Connes & Marcolli, which this post is following) is to describe the algebra in terms of functions on $\mathbb{Q}$-lattices, as this approach is the one used in generalizations. We'll denote this subalgebra by $\mathcal{A}_{1,\mathbb{Q}}$.
The description here is actually quite similar to modular forms: a function $f$ on the space of 1dQL is homogeneous of weight $k$ if it satisfies $f(\lambda(\Lambda, \phi)) = \lambda^{-k} f(\Lambda,\phi)$. E.g., the space of functions on 1dQLs of weight zero is $C(\hat{\mathbb{Z}})$. For $a \in \mathbb{Q}/\mathbb{Z}$, we can define a class of functions on 1dQLs of weight zero by:

$\displaystyle e_{1,a}(\Lambda,\phi) = c(\Lambda) \sum_{y \in \Lambda + \phi(a)} y^{-1$

When $\phi(a) \neq 0$. $e_{1,a}(\Lambda,\phi) = 0$ for $\phi(a) = 0$ or $\phi(a) \in \Lambda$. Also, $c(\Lambda) = \frac{1}{2\pi i} | \Lambda |$, $|\Lambda|$ the covolume of the lattice. These $e_{1,a}$'s are the key functions.

Thrm 1 (Connes & Marcolli 3.30) The functions $e_{1,a}$ have the following properties:

1. The $e_{1,a}$ for $a \in \mathbb{Q}/\mathbb{Z}$ generate the group ring $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$.
2. $\mathcal{A}_{1,\mathbb{Q}}$ is generated by the $e_{1,a}$ and the $\mu_n$ and $\mu_n^{\ast}$ from the previous post.
3. The complexification of $\mathcal{A}_{1,\mathbb{Q}}$ is the $C^{*}$-algebra $\mathcal{A}_1$, generated by the above elements.
   

The proof relies on relations between the $e_{1,a}$ and their generalizations (I do not fully understand the details), and is related to similar work on Eisenstien series (Weil, Elliptic Functions according to Eisenstein and Kronecker contains similar work). While I cannot prove much of anything here, I hope to describe those relations and give a sketch of their proofs.
We can generalize these functions as follows: for $a \in \mathbb{Q}/\mathbb{Z}$, define:

$\displaystyle \epsilon_{k,a}(\Lambda, \phi) = \sum_{y \in \Lambda + \phi(a)} y^{-k$

When $\phi(a) \neq 0$, otherwise $\epsilon_{k,a}(\Lambda, \phi) = \lambda_k c(\Lambda)^{-k}$, with $\lambda_k = (2^k -1)\gamma_k$, and $\gamma_k$ defined by the generating series:

$\displaystyle \frac{1}{e^x - 1} = 1 - \frac{x}{2} - \sum_{j=1}^{\infty} \gamma_{2j}x^{2j$

The $\epsilon_{k,a}$ are of weight $k$. We define the weight $0$ functions by $e_{k,a} = c^k \epsilon_{k,a}$.
The functions $e_{k,a}$ can actually be expressed as polynomials of $e_{1,a}$. For instance, if you let $p_1(u)=u$ and $p_{k+1}(u) = \frac{1}{k}(u^2 - \frac{1}{4}) p^{\prime}_k (u)$, then $e_{k,a} = p_k(e_{1,a})$. The proof of this statement apparently follows from basic formulas for the trigonometric analog of the Eisenstein series. Connes & Marcolli cite the book by Weil, ch2, but I've yet to digest that material (or the proof that they give).
Our $e_{k,a}$ can also be expressed in terms of our generators $e(r)$ of $\mathcal{A}_1$. First, consider them as functions on 1dQLs by $e(r)(\Lambda,\phi) = \text{exp}(2 \pi i \rho(r))$ where $(\Lambda,\phi) = \lambda(\mathbb{Z},\rho)$. For $a \in \mathbb{Q}/\mathbb{Z}$ and $n > 0$ such that $na = 0$, we have the following relation:

$\displaystyle e_{1,a} = \sum_{k=1}^{n-1}\left( \frac{k}{n} - \frac{1}{2} \right) e(ka) $

I have one more relation I'd like to state, but before I can I need to define a new concept. Let $\pi_n$ by the characteristic function of the set of all 1dQL's $(\Lambda,\phi)$ such that the restriction of the map $\phi$ to the $n$-torsion points of $\mathbb{Q}/\mathbb{Z}$, $\phi_n$, is 0. (If this is the case, we say that the $\mathbb{Q}$-lattice $(\Lambda,\phi)$ is divisible by $n$.) Now for a given $N > 0$, the functions $e_{k,a}$ satisfy:

$\displaystyle \sum_{Na = 0} e_{k,a} = \gamma_k \sum_{d | N} C_{k,d} \pi_d $

Where

$\displaystyle C_{k,d} = (2^k - 2)f_1(d) + N^k f_{1-k}(d)$

$f_1$ the Euler totient function, and

$\displaystyle f_k (n) = n^k \, \prod_{p \, \text{prime}, \, p|n} (1-p^{-k})$

The proof of the main theorem relies on these 3 relations. If we let $\mathcal{B}_{\mathbb{Q}}$ be the algebra generated over $\mathbb{Q}$ by the $e_{1,a}$, then our polynomial relation says that $e_{k,a} \in \mathcal{B}_{\mathbb{Q}}$. Our relation with the $e(r)$ says that $\mathcal{B}_{\mathbb{Q}}$ is a subalgebra of the group ring $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$. We use that last relation to show that our characteristic functions $\pi_n$ belong to $\mathcal{B}_{\mathbb{Q}}$, and then to show that all $e(r)$ are in $\mathcal{B}_{\mathbb{Q}}$ as well, so that $\mathcal{B}_{\mathbb{Q}} = \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$. The second and third parts of the theorem follow easily from that.
This section has a lot of details with Eisenstein series, et cetera, that I did not work out. As this section is more to do with how to generalize the Bost-Connes system, I didn't want to spend too much time on it; it may be tangential to my goals for my project. Though, I may come back to it later on. In the meantime, my next post will be describing the interaction with class field theory.