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.