The Étale Groupoids of the Bost-Connes System

PDF version of the past 3 posts.

Recall that a 1-dimensional $ {\mathbb{Q}}$-lattice (henceforth called a 1dQL) can be denoted by $ {(\frac{1}{\lambda} \mathbb{Z}, \frac{1}{\lambda} \rho)}$ with $ {\lambda}$ a positive real and $ {\rho \in End(\mathbb{Q}/\mathbb{Z})=\hat{\mathbb{Z}}}$. Let $ {\mathcal{Q}_1}$ denote the set of commensurability relations of such lattices. Then $ {\mathcal{Q}_1}$ consists of ordered pairs $ {(\Lambda_1,\Lambda_2)}$ of commensurable 1dQLs. There's a natural composition of elements, namely:

$ {(\Lambda_1,\Lambda_2) \circ (\Lambda_3, \Lambda_4) = (\Lambda_1,\Lambda_4)}$ defined for $ {\Lambda_2=\Lambda_3}$.

It's not too hard to see that each element has an inverse. Indeed $ {(\Lambda_1,\Lambda_2)^{-1} = (\Lambda_2,\Lambda_1)}$, so that $ {(\Lambda_2,\Lambda_1) \circ (\Lambda_1,\Lambda_4) = (\Lambda_3, \Lambda_4)}$. Hence we have a groupoid. The units $ {\mathcal{Q}_1^0}$ consists of the pairs $ {(\Lambda, \Lambda)}$, id est, just the set of 1dQLs.
We can actually give a much more clear description of this groupoid. Let $ {\mathcal{G}_1 = \{(r,\rho,\lambda) : r \in \mathbb{Q}^{*}_{+}, \, \rho \in \hat{\mathbb{Z}}, \, \lambda \in \mathbb{R}^{*}_{+} \; such \; that \; r \rho \in \hat{\mathbb{Z}} \}}$ be the groupoid with the composition

$ {(r_1,\rho_1,\lambda_1) \circ (r_2,\rho_2,\lambda_2) = (r_1 r_2, \rho_2, \lambda_2)}$ defined for $ {r_2 \rho_2 = \rho_1}$, $ {r_2 \lambda_2 = \lambda_1}$.

and inverse elements

$ \displaystyle (r,\rho,\lambda)^{-1} = \left(\frac{1}{r},r \rho, r \lambda \right) $

Proposition. The map $ {\phi : \mathcal{G}_1 \rightarrow \mathcal{Q}_1}$ by

$ \displaystyle (r,\rho,\lambda) \mapsto \left( \left( \frac{1}{r \lambda} \mathbb{Z}, \frac{1}{\lambda} \rho \right), \left( \frac{1}{\lambda} \mathbb{Z}, \frac{1}{\lambda} \rho \right) \right)$

is a groupoid isomorphism. (Connes, Marcolli; Noncommutative Geometry, Quantum Fields, and Motives. Lemma 3.21-2.)
Proof: First note that the two 1dQLs, $ {\left( \frac{1}{r \lambda} \mathbb{Z}, \frac{1}{\lambda} \rho \right)}$ and $ {\left( \frac{1}{\lambda} \mathbb{Z}, \frac{1}{\lambda} \rho \right)}$ are commensurable. Indeed, $ {\mathbb{Q} \frac{1}{r \lambda} \mathbb{Z} = \frac{1}{\lambda} \mathbb{Q} \mathbb{Z}}$.

Second, note that every pair of commensurable 1dQLs is of this form. To see this, let $ {(\frac{1}{\lambda_1} \mathbb{Z}, \frac{1}{\lambda_1} \rho_1)}$ and $ {(\frac{1}{\lambda_2} \mathbb{Z}, \frac{1}{\lambda_2} \rho_2)}$ be another pair of commensurable 1dQLs. $ {\frac{1}{\lambda_2}}$ then must be a rational multiple of $ {\frac{1}{\lambda_1}}$, id est, $ {\lambda_1 = r \lambda_2}$ for a strictly positive rational $ {r=\frac{a}{b}}$ (see previous posts on $ {\mathbb{Q}}$-lattices). Additionally, we must have $ {\frac{1}{\lambda_1} \rho_1 = \frac{1}{\lambda_2} \rho_2 \mod (\frac{1}{\lambda_1} \mathbb{Z} + \frac{1}{\lambda_2} \mathbb{Z})}$ Factoring out by $ {\lambda_2}$, this yields $ {\frac{1}{r} \rho_1 = \rho_2 \mod \frac{1}{a} \mathbb{Z}}$, and then $ {a\rho_2 - b \rho_1 = 0}$ and finally $ {\rho_1 = r \rho_2}$, so that $ {\frac{1}{\lambda_1}\rho_1=\frac{1}{r \lambda_2} r \rho_2 = \frac{1}{\lambda_2}\rho_2}$. Hence the map is surjective.

Finally, note that the map preserves the groupoid composition and inversion. The composition

$ \displaystyle \left( (\frac{1}{r_1 \lambda_1} \mathbb{Z}, \frac{1}{\lambda_1} \rho_1),(\frac{1}{\lambda_1} \mathbb{Z}, \frac{1}{\lambda_1} \rho_1) \right) \circ \left( (\frac{1}{r_2 \lambda_2} \mathbb{Z}, \frac{1}{\lambda_2} \rho_2),(\frac{1}{\lambda_2} \mathbb{Z}, \frac{1}{\lambda_2} \rho_2) \right)$

is only defined in $ {\mathcal{Q}_1}$ if $ {\lambda_1=r_2 \lambda_2}$ and $ {\rho_1 = r_2 \rho_2}$. In that case, it equals

$ \displaystyle \left( (\frac{1}{r_1 \lambda_1} \mathbb{Z}, \frac{1}{\lambda_1} \rho_1),(\frac{1}{\lambda_2} \mathbb{Z}, \frac{1}{\lambda_2} \rho_2) \right)=\left( (\frac{1}{r_1 r_2 \lambda_2} \mathbb{Z}, \frac{1}{\lambda_2} \rho_2),(\frac{1}{\lambda_2} \mathbb{Z}, \frac{1}{\lambda_2} \rho_2) \right) $

Which is exactly $ {\phi(r_1 r_2, \rho_2,\lambda_2)}$. Additionally,

$ \displaystyle \phi(\frac{1}{r},r \rho, r \lambda ) = \left( (\frac{1}{\lambda} \mathbb{Z}, \frac{1}{\lambda} \rho), (\frac{1}{r \lambda} \mathbb{Z}, \frac{1}{\lambda} \rho) \right)$

Which is the inverse of $ {\left( (\frac{1}{r \lambda} \mathbb{Z}, \frac{1}{\lambda} \rho), (\frac{1}{\lambda} \mathbb{Z}, \frac{1}{\lambda} \rho) \right)}$ $ \Box$


Proposition. $ {\mathcal{G}_1}$ is an Étale groupoid. (I do not yet know the significance of this statement.)

Proof: Since $ {\mathcal{G}_1}$ has the product topology, $ {\mathbb{Q}_{+}^{*}}$ has the subspace topology, multiplication in all spaces is continuous, and division in $ {\mathbb{Q}_{+}^{*}}$ is continuous, the groupoid operations are homeomorphisms. $ \Box$


But we're not done yet. All the action in the Bost-Connes system comes not from $ {\mathcal{G}_1}$, but from the commensurability relations of 1dQLs modulo scaling, id est, from $ {\mathcal{Q}_1/\mathbb{R}_{+}^{*}}$. The unit set $ {(\mathcal{Q}_1/\mathbb{R}_{+}^{*})^0}$ is simply $ {\hat{\mathbb{Z}}}$ in this case. As you might expect, we can model this groupoid with an Étale groupoid similar to $ {\mathcal{G}_1}$: Let

$ \displaystyle \mathcal{U}_1 = \{ (r,\rho) : r \in \mathbb{Q}_{+}^{*})^0, \rho \in \hat{\mathbb{Z}} \; such \; that \; r \rho \in \hat{\mathbb{Z}}\}$

With composition

$ { (r_1,\rho_1) \circ (r_2,\rho_2) = (r_1 r_2,\rho_2)}$ defined when $ {\rho_1 = r_2 \rho_2}$.

.
Proposition. The map $ {\gamma : \mathcal{U}_1 \rightarrow \mathcal{Q}_1/\mathbb{R}_{+}^{*}}$ by

$ \displaystyle (r,\rho) \mapsto \left( (\frac{1}{r}\mathbb{Z},\rho), (\mathbb{Z},\rho) \right)$

is a groupoid isomorphism. (Connes, Marcolli; Noncommutative Geometry, Quantum Fields, and Motives. Prop 3.22.)
Proof: The proof of this is similar to the last one, so I won't repeat myself. Oddly enough, Connes & Marcolli give a short proof of the first isomorphism, and a long proof of this one. $ \Box$

Since the topology of the groupoids largely comes from $ {\mathbb{R}}$, they are also locally compact (at least, I think that's the reason.)

1. Why This Matters

It's important to keep the larger goal of the project in mind when working out these smaller details. My goal is to describe the construction of the Bost-Connes system, as I did here. Connes & Marcolli give an explicit description in terms of generators and relations of the Bost-Connes algebra in Noncommutative Geometry, Quantum Fields, Motives [1]. Additionally, Marcolli gives the same description in Lectures on Arithmetic Noncommutative Geometry [2], Connes & Marcolli give it again in $ {\mathbb{Q}}$-Lattices: Quantum Statistical Mechanics and Galois Theory [3], and Bost & Connes give it in Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory [4]. However, only [1] and [4] provide a proof for the description, and the latter only in terms of Hecke Algebras. [1], however, starts from convolution algebras of these Étale groupoids, and shows the relation with the Hecke algebra description in the paper by Bost & Connes [4]. Hence, it's important that I understand and am able to describe these groupoids.

2. What's next

The next step is a careful study of the convolution and C* groupoid algebras of the above structures, leading to a proof of the explicit description of the Bost-Connes system (Prop. 3.23 in Noncommutative Geometry, Quantum Fields, Motives). Following that, I'll need to understand the relation with the Hecke Algebras given by Bost & Connes and the symmetries of the system. Then I can begin a study of the class field theory of $ {\mathbb{Q}}$ and learn how to apply the Bost-Connes system.

Eventually, I'll study the two other Quantum Statistical Mechanical systems, but first I need to finish the Bost-Connes system and its applications, and learn its corresponding sub algebras and Shimura varieties, which will be vital to generalizing the techniques used to apply it.

3. Sources

Noncommutative Geometry, Quantum Fields, Motives. Connes, Marcolli
Lectures on Arithmetic Noncommutative Geometry. Marcolli (the usual two)
A Homology Theory for Étale groupoids. Marius Crainic and Ieke Moerdijk
Abstract Algebra. Dummit, Foote (appendix on category theory.)
Category Theory. Steve Awodey
Wikipedia on Groupoids