Sunday, January 30, 2011

The Global Artin Map for $\mathbb{Q}$

I'm discussing the global Artin homomorphism $\theta : \mathcal{C}_\mathbb{Q} \rightarrow \text{Gal}(\mathbb{Q}^\text{ab}/\mathbb{Q})$ used in our key statement about the Bost-Connes system, as well as the Adeles, Ideles, and other components needed to consider it.

In my last significant post I stated the global Artin map for the rationals intertwined with the Galois action on the values of Bost-Connes KMS states. I wanted to talk about the Artin map in a bit more detail. The general case for any number field $\mathbb{K}$ is a bit too complicated for me to discuss right now, but the case for $\mathbb{Q}$ isn't bad at all. Let's start with $\mathcal{C}_\mathbb{Q} = \mathbb{A}^{\ast}_\mathbb{Q}/\mathbb{Q}^{\ast}$, the Idele class group of $\mathbb{Q}$.
First recall the definition of the Adeles, $\mathbb{A}_\mathbb{Q}$.

$ \displaystyle \mathbb{A}_\mathbb{Q} = \mathbb{R} \times \prod_{p \; \text{prime}}^{\prime} \mathbb{Q}_p $

Where $\mathbb{Q}_p$ is the completed field of $p$-adic numbers. The idea here is that we're looking at all possible completions of the number field $\mathbb{Q}$, including the standard absolute value and all $p$-adic valuations. Each of these completed fields is a ``local'' field, containing information about $\mathbb{Q}$ at a particular prime, and the ring of Adeles combines all that information into one giant ring. The $\prime$ on the product $\prod^{\prime}$ indicates that this product is restricted over $\mathbb{Z}_p$, id est, it has the condition that

$ \displaystyle \mathbb{R} \times \prod_{p \; \text{prime}}^{\prime} \mathbb{Q}_p = \{ a=(a_\infty,a_2,a_3,\ldots) \in \mathbb{A}_\mathbb{Q} \; \text{iff} \; a_p \in \mathbb{Z}_p \; \text{for all but finitely many p}\$

$\mathbb{A}^{\ast}_\mathbb{Q}$ is the set of all invertible elements in the ring, we can write it $\mathbb{R}^\ast \times \prod_p^\prime \mathbb{Q}_p^\ast$. As $\mathbb{Q}_p$ is the field of fractions for $\mathbb{Z}_p$, we have an isomorphism $\mathbb{Q}_p \cong \mathbb{Z}_p[\frac{1}{p}] \cong p^\mathbb{Z} \times \mathbb{Z}_p$. If we identify $p^\mathbb{Z}$ with $\mathbb{Z}$ we can write $\mathbb{Q}_p^\ast \cong \mathbb{Z} \times \mathbb{Z}^\ast_p$ and

$ \displaystyle \mathbb{A}_\mathbb{Q}^\ast = \mathbb{R}^\ast \times \prod_{p \; \text{prime}}^{\prime} \mathbb{Q}_p^\ast \cong \{\pm1\} \times \mathbb{R}_{>0} \times \prod \mathbb{Z}_p^\ast \times \bigoplus_p \mathbb{Z}$

Now for an $r \in \mathbb{Q}^\ast$, we can write $r = \text{sgn}(r) \prod_p p^n(p)$, so we can take $\mathbb{Q}^\ast$ to $\{\pm1\} \times \bigoplus_p \mathbb{Z}$ by $r \mapsto (\text{sgn}(r),n(p)_p)$. By noted that $\mathbb{R}_{>0} \cong \mathbb{R}$ by logarithm, we can write:

$ \displaystyle \mathbb{A}_\mathbb{Q}^\ast \cong \mathbb{Q}^\ast \times \mathbb{R} \times \prod_p \mathbb{Z}_p^\ast$

Now recall our favourite profinite group $\hat{\mathbb{Z}} = \varprojlim_k \mathbb{Z}/k\mathbb{Z} \cong \text{End}(\mathbb{Q}/\mathbb{Z})$. Since we can write each $k$ as $p_1^{n(p_1)} \ldots p_l^{n(p_l)}$, we also have

$ \displaystyle \mathbb{Z}/k\mathbb{Z} = \prod_{i=1}^{l} \mathbb{Z}/p_i^{n(p_i)} \mathbb{Z}$

After passing to the inverse limit we have:

$ \displaystyle \hat{\mathbb{Z}} = \varprojlim_k \mathbb{Z}/k\mathbb{Z} = \prod_p \varprojlim_n \mathbb{Z}/p^n\mathbb{Z} = \prod_p \mathbb{Z}_p$

Thus $ \mathbb{A}_\mathbb{Q}^\ast \cong \mathbb{Q}^\ast \times \mathbb{R} \times \hat{\mathbb{Z}}^\ast $ and finally:

$ \displaystyle \mathcal{C}_\mathbb{Q} \cong \mathbb{R} \times \hat{\mathbb{Z}}^\ast$

Now I cannot prove it, but class field theory tells us that the map $\theta:\mathcal{C}_\mathbb{Q} \rightarrow \text{Gal}(\mathbb{Q}^\text{ab}/\mathbb{Q})$ is surjective, and the kernel is the connected component of the identity on $\mathcal{C}_\mathbb{Q}$, which we denote $\mathcal{D}_\mathbb{Q} = \mathbb{R}$. Hence $\mathcal{C}_\mathbb{Q} / \mathcal{D}_\mathbb{Q} \cong \hat{\mathbb{Z}}^\ast$ and we have an isomorphism:

$ \displaystyle \theta : \hat{\mathbb{Z}}^\ast \rightarrow \text{Gal}(\mathbb{Q}^\text{ab}/\mathbb{Q})$

We can see that these two groups are isomorphic assuming only the Kronecker-Weber theorem (which says that $\mathbb{Q}^\text{ab} = \mathbb{Q}^\text{cycl}$) and some facts about inverse limits and Galois theory. $\mathbb{Q}^\text{cycl} = \bigcup_n \mathbb{Q}(\zeta_n)$, where $\zeta_n$ is the n-th root of unity. Recall that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\ast$. Hence we have

$ \displaystyle \text{Gal}(\mathbb{Q}^\text{cycl}/\mathbb{Q}) = \varprojlim_n \text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong \varprojlim_n (\mathbb{Z}/n\mathbb{Z})^\ast = \hat{\mathbb{Z}}^\ast$

Thanks to the following sources for help on this post:

Friday, January 14, 2011

Thoughts on a QSM proof the Kronecker-Weber Theorem

As I discussed in my last posts, our whole interest in Bost-Connes systems is due to its interaction with the class field theory of certain number fields. In the case of $\mathbb{Q}$, which I'm covering in my MSci project, we can recover the full class field theory and get a description of the maximal abelian/cyclotomic extension of $\mathbb{Q}$ without reference to field extensions. To be precise, we have

Thrm 1 (Prop 3.33 Connes & Marcolli) The quotient $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]/(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J)$ is a field isomorphic to $\mathbb{Q}^\text{cycl}$ and the ideal $(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J)$ in $C^\ast(\mathbb{Q}/\mathbb{Z})$ is equal to ideal generated by the $\pi_m$ in $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$, where $J$ and $\pi_m$ are defined in my last post.

Despite this result, Connes & Marcolli state that its still an open problem to find a proof the of the Kronecker-Weber theorem using methods from Quantum Statistical Mechanics. After some googling today, I couldn't find any more information on this, and [perhaps due to my ignorance] I can't figure out why it would be so difficult. I must not understand something. But it would seem to be that given a finite abelian extention of the rationals $\mathbb{K}$ our task would be to find an ideal $K$ such that $(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J) \subset K$ and so that $\mathbb{K} \sim \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]/K \subset \mathbb{Q}[\mathbb{Q}/\mathbb{Z}]/(\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J) \sim \mathbb{Q}^\text{cycl}$. I obviously don't know much about the details of implementing it, but it might be something to think about as I study the proof of the that main class field theory result of the Bost Connes system. Any thoughts?

Wednesday, January 12, 2011

If you ask me where I’m going, I’ll tell you where I’ve been.

This post is a brief recap of my work on my MSci project at UCL since July 2010. This post was hastily written. Please excuse any grammatical errors or typos.

Since I've now stated the significance of the Bost-Connes system, its time to consider the next step. But it's now mid-January, and due-dates are fast approaching for my MSci project. Thus, I thought I would take some time to write a short summary of everything I've covered since I started in July. I've actually not done much, and I'm ashamed of how little work I managed to do in such a long time. In any case, lets go over where I've been:

  • Quantum Statistical Mechanics
    Basic definitions (no examples), KMS states, symmetry groups. There's a lot of details I missed, particularly about the relationship between GNS representations and extremal KMS states. I have books by Bratteli and Robinson on Operator Algebras and Quantum Statistical Mechanics and Haag's Local Quantum Physics to help me out there.

  • $\mathbb{Q}$-lattices and Commensurability
    Basic definitions, and conversations with Professor Kim to help build some intuition, inversion and division.

  • $C^\ast$-algebras
    Basic definitions and few examples, though I'm missing quite a few details that I should like, particularly GNS representations. (Gelfand-Naimark Theorem would be nice, but not really needed here.)

  • Profinite Completion of $\mathbb{Z}$
    I spent quite a few hours trying to figure out Pontryagin duality and how $\text{Hom}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$. I would like to go over this again, using more sophisticated methods (category theory) and better understand how $C(\hat{\mathbb{Z}}) = C^\ast(\mathbb{Q}/\mathbb{Z})$

  • Category Theory, Groupoids, and \'{E}tale Groupids
    Categories, Functors, Groupoids, a few examples, \'{E}tale Groupids (a topology imposed on a groupoid), which I don't think I fully understand and would like to review.

  • Convolution Algebras and $C^\ast$ groupoid algebras for an \'{E}tale Groupid
    Basic definition on convolution, general discussion of the $C^\ast$ completion of a group/oid algebra via a Haar measure (without rigorously defining the Haar measure), discussion of why an Etale Groupid does not need this, and the completion of a $C^\ast$ groupoid algebra this way via a maximal norm of the image of a representation onto a Hilbert space.

  • Generators and Relations for Bost Connes System via semigroup crossed products
    Proof that the $C^\ast$ groupoid algebra for the Bost Connes System is a semigroup crossed product (I don't yet understand the proof fully, despite conversations with Dr Javier Lopez), which is vitally important, because later research shows that this is the ``right'' way to describe Bost-Connes type systems. Proof of the generators and relations via the semigroup crossed product.

  • (Briefly) Hecke Algebras and the Bost Connes System
    Brief definition and description of the Bost Connes system via Hecke Algebras, as they were used in the original paper.

  • (Briefly) The Arithmetic Subalgebra and its analysis inspired from Eisenstein Series
    I didn't spend too much time on this, because I \underline{think} it's more important for generalizations of the Bost-Connes system then for the 1-dim case, and I don't plan to discuss the generalizations in detail. In any case, I went over the results of trigonometric analogue of the Eisenstein series to study the arithmetic subalgebra of the Bost-Connes system. Its easiest to see this algebra exist from the Hecke algebra perspective (I think). Weil's boook on Elliptic Functions contains a lot of the ideas needed to do this in detail.

  • Class Field Theory and the action of the KMS states of the Bost Connes System
    Very brief discussion of adeles, ideles, and the global Artin map, and statement about how the Bost Connes System can recover a presentation of the maximal abelian extension of $\mathbb{Q}$ without reference to field extensions. Plan to generalize this result. See my last post, as well my notes from my October talk at the Undergrad Maths Colloquium

So what's next? Besides filling in the details for the above (I'm embarrassed that despite taking so long to cover that, I glossed over a lot of details), I essentially have two items left I want to cover: 1) the proof of the theorem in my last post and 2) a summary of the on-going work on generalizations of the BC system. The latter may require learning about Shimura varieties. Both will certainly require a better understanding of class field theory: the adeles, ideles and Artin maps. So that is where I'm going!

Class Field Theory & the Bost-Connes System

In this post I describe the classification of the KMS states of the Bost-Connes system and their relationship to the class field theory of $\mathbb{Q}$, additionally, we show how the rational subalgebra gives us a presentation of $\mathbb{Q}^\text{cycl}$ without any reference to field extensions. Finally, we describe the ``plan'' for generalizing these methods to an abitrary number field.

In all this talk about groupoids and algebras, one may have forgotten that the Bost Connes system $\mathcal{A}_1 = C^{\ast}(\mathcal{G}_1/\mathbb{R}_{+})$ (with $\mathcal{G}_1$ being the groupoid of the commensurability classes of 1dQLs) is a Quantum Statistical Mechanical system with time evolution $\sigma_t$. The ``proper'' way to study the QSM system $(\mathcal{A}_1,\sigma_t)$ is to look at its KMS states. As oft stated in my blog, the KMS states intertwine with the Galois group of the maximal Abelian extension of $\mathbb{Q}$. We can finally state this result properly, but first I need to pull a few quick ideas from Class Field Theory (ideas that I don't understand properly myself). First we want to define the ring of adeles $\mathbb{A}_\mathbb{Q}$ of $\mathbb{Q}$ (I'll define this for a general number field in another blog).
Consider the $p$-adic integers, $\mathbb{Z}_p$, which is the inverse limit of $\mathbb{Z}/p^n\mathbb{Z}$. Each $\mathbb{Z}/p^n\mathbb{Z}$ has the discrete topology, so the Tychonov's theorem says that $\mathbb{Z}_p$ has a compact topology, it also has field of fractions $\mathbb{Q}_p$. The adeles $\mathbb{A}_\mathbb{Q}$ are the restricted product of $\mathbb{Q}_p$ over $\mathbb{Z}_p$ for all primes $p$, where ``restricted'' means that elements of $\mathbb{A}_\mathbb{Q}$ are the infinite sequences $a = (a_2, a_3, \ldots)$ such that all by finitely many $a_p \in \mathbb{Z}_p$.
The group of Id\'{e}les is the invertible elements $\mathbb{A}^{\ast}_\mathbb{Q}$. But this isn't enough for us. We need the id\'{e}le class group $\mathcal{C}_\mathbb{Q} = \mathbb{A}^{\ast}_\mathbb{Q} / \mathbb{Q}$. Class Field/Algebraic Number Theory says that we have a group homomorphism, the global Artin homomorphism:

$\displaystyle \theta : \mathcal{C}_\mathbb{Q} \rightarrow \text{Gal}(\mathbb{Q}^\text{cycl}/\mathbb{Q})$

This homomorphism is surjective, and its kernel is a connected component of the identity in $\mathcal{C}_\mathbb{Q}$, which we'll denote $\mathcal{D}_\mathbb{Q}$. This means we have an isomorphism. For $\mathbb{Q}$, it turns out that $\mathcal{C}_\mathbb{Q} / \mathcal{D}_\mathbb{Q} = \hat{\mathbb{Z}}^{\ast}$, so we have the following isomorphism:

$\displaystyle \theta: \hat{\mathbb{Z}}^{\ast} \rightarrow \text{Gal}(\mathbb{Q}^\text{cycl}/\mathbb{Q})$

One should note that I don't understand any of the above class field theory, and I've only looked up enough details to be able to state the isomorphism. We're now ready to state the classification of the KMS states and the main result of the Bost-Connes system (not to mention my project!) This is Theorem 3.32 in the monograph by Connes and Marcolli, first stated in Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory by Bost and Connes. Recall that that $\xi_\beta$ denotes the set of extremal KMS states at inverse temperature $\beta$.

Thrm 1 The KMS states for the Quantum Stastical Mechanical System $(\mathcal{A}_1,\sigma_t)$ behave as follows:

  • $\xi_\beta$ is a singleton for all $0 < \beta \leq 1$. The KMS state, when restricted to the subalgebra $\mathcal{B}_\mathbb{Q}$ has values:

    $\displaystyle \phi_\beta(e(a/b)) = b^{-\beta} \prod_{p \; \text{prime},\; p|b} \left( \frac{1-p^{\beta-1}}{1-p^{-1}} \right)$

  • For $1 < \beta \leq \infty$, the elements of $\xi_\beta$ are indexed by invertible 1dQLs modulo scaling $\rho \in \hat{\mathbb{Q}}$. On $\mathcal{B}_\mathbb{Q}$ they take the values ($\zeta_{a/b}$ being roots of unity):

    $\displaystyle \phi_{\beta,\rho}(e(a\b)) = \frac{1}{\zeta(\beta)} \sum_{n=1}^{\infty} n^{-\beta} \rho ( \zeta_{a/b}^n)$

    Hence the partition function of the system is the Riemann zeta function. (I do not understand the argument or its significance.)
  • The extreme zero-temperature KMS states $\xi_\infty$ have the property that:

    $\displaystyle \phi(\mathcal{A}_{1,\mathbb{Q}}) \subset \mathbb{Q}^{\text{cycl}} \; \text{for all}\; \phi \in \xi_\infty$

    Moreover, the global Artin isomorphim interacts with the Galois action on the values of the KMS state as follows:

    $\displaystyle \gamma(\phi(f)) = \phi ( \theta^{-1}(\gamma) f)$

A proof of that theorem is not given in the monograph (but it is found elsewhere, at the least in terms of Hecke algebras in the aforementioned paper!) I do intend to cover the proof in my project. Additionally we can actual use this idea to give a presentation of $\mathbb{Q}^\text{cycl}$. Consider $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \subset C^\ast(\mathbb{Q}/\mathbb{Z}) \sim C(\hat{\mathbb{Z}})$.
If you let $u(r)$ be the canonical basis of $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$, and let

$\displaystyle \pi_m = \frac{1}{m} \sum_{r \in \mathbb{Q}/\mathbb{Z},\; mr=0} u(r)$

For $m > 1$ be idempotents. Finally, let $J \subset C^\ast(\mathbb{Q}/\mathbb{Z})$ by the ideal generated by the $\pi_m$. I won't prove it here, but it turns out that $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J$ is equal to the ideal generated by the $\pi_m$ in $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}]$. Moreover, $\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] / (\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] \cap J)$ is a field isomorphic to $\mathbb{Q}^\text{cycl}$
Connes and Marcolli give a ``plan of action'' for extending this result to study the class field theory of a general number field $\mathbb{K}$. I give it below:
  1. For $\mathbb{K}$, construct a quantum statistical mechanical system $(\mathcal{A}_\mathbb{K},\sigma_t)$ that has the Dedekind zeta function $\zeta_\mathbb{K}(\beta)$ as a partition function, and whose symmetries induce an action of $\mathcal{C}_\mathbb{K}/\mathcal{D}_\mathbb{K}$ on the KMS states.
  2. Find an arithmetic subalgebra $\mathcal{A}^\text{arithm}_\mathbb{K}$ that has the aforementioned interaction with the class id\'{e}le group and global Artin isomorphism: $\phi \circ \alpha(f) = \sigma(\alpha)\phi(f)$
  3. Compute the presentation of the algebra $\mathcal{A}^\text{arithm}_\mathbb{K}$ and the extremal zero-temperature KMS states in terms of their values on $\mathcal{A}^\text{arithm}_\mathbb{K}$.

In addition to proving the above theorem, I also hope to discuss (in a high level) the recent work on implementing that plan.

Saturday, January 1, 2011

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 $


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

$f_1$ 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.