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*

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