Connes and Marcolli's monographs define a $\mathbb{Q}$-lattice as a lattice $\Lambda \in \mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$. We call two $\mathbb{Q}$-lattices $\Lambda_1$ and $\Lambda_2$ commensurable ($\Lambda_1 \sim \Lambda_2$) if:

1) $\mathbb{Q} \Lambda_1 =\mathbb{Q} \Lambda_2$ and

2) $\phi_1$ = $\phi_2$ mod $\Lambda_1 + \Lambda_2$.

Condition 1) means that there exists some $\mathbb{Q}$-span in $\mathbb{R}^n$ such that $\Lambda_1$ and $\Lambda_2$ both sit inside it. For instance, both $\frac{1}{2} \mathbb{Z}$ and $\frac{1}{5} \mathbb{Z}$ sit inside $\mathbb{Q}$, so they meet condition 1. But if we take $< (1,0), (0,\sqrt{2}) >$ and $<(\frac{1}{2},0), (0,0) >$ we cannot find any vectors in $\mathbb{R}^2$ that, when spanned over $\mathbb{Q}$, would contain both of those lattices.

Condition 2) is actually how I originally understood it. The homomorphism must "label the same points". More formally, Lets say $X =\mathbb{Q} \Lambda_1 =\mathbb{Q} \Lambda_2$. For each $\Lambda_i$ we have a projection map $\pi_i : X/\Lambda_i \to X/(\Lambda_1 + \Lambda_2)$ by $r+\Lambda_i \mapsto r + (\Lambda_1 + \Lambda_2)$ (of course, $r$ may not be the best representative element in $\Lambda_1 + \Lambda_2$). Condition two says that $\pi_1 \circ \phi_1 = \pi_2 \circ \phi_2$.

**1-dim Q-lattices**

We can describe the Bost-Connes system with $\mathbb{Q}$-lattices in $\mathbb{R}$, and 1-dim $\mathbb{Q}$-lattices can be described by $\hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}$, where $\hat{\mathbb{Z}} = \varprojlim_{n} \mathbb{Z} / n\mathbb{Z}$. First note any lattice $\Lambda \in \mathbb{R}$ can be described by $\lambda \mathbb{Z}$, where $\lambda \in \mathbb{R}_{+}^{*}$. So the only data we need is the homomorphism $\phi : \mathbb{Q} / \mathbb{Z} \to \mathbb{Q} / \lambda \mathbb{Z}$. But apparently $\hat{\mathbb{Z}} \cong Hom (\mathbb{Q} / \mathbb{Z},\mathbb{Q} / \mathbb{Z})$, that is, $\hat{\mathbb{Z}}$ is isomorphic to the set of all endomorphisms of $\mathbb{Q} / \mathbb{Z}$ (I'm not yet sure how! Something to do with Pontryagin duality?), so all that remains is a choice of $\rho \in \hat{\mathbb{Z}}$ and we have $(\Lambda,\phi) \mapsto (\rho,\lambda)$. One should note that the inverse of this map is well defined, so we can identify the set of all 1-dim $\mathbb{Q}$-lattices with $\hat{\mathbb{Z}} \times \mathbb{R}_{+}^{*}$.

I'll continue reading about 1-dim $\mathbb{Q}$-lattices for the next several days, though I do still plan to take some time to catch up on some algebra (algebras over a field, group rings, group algebras, C* algebras, their representations, group C* algebras, Pontryagin duels, etale groupoids.)

Thanks to Professor Kim and the folks at mathoverlow for helping me understand this.

Also, for the next few days I will all but abandon Marcolli's monograph; that text goes straight into a description of the Bost-Connes system via Shimura varieties, which is a bit above my level. I do intent to understand the Shimura variety and the Hecke algebra description of Bost-Connes eventually, however.

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