English translation of M.-F. Vigneras "Arithmétique des algèbres de quaternions"

Question

I am looking to understand a citation about the connection of quaternion algebra over number fields which when embedded into $\mathbb{C}$, leads to a discrete subgroup of $SL_2(\mathbb{C})$ which causes tilings of the hyperbolic 3-manifold $\mathbb{H}^3$. The author mentions the chapter 4 in the French book M.-F. Vigneras -"Arithmétique des algèbres de quaternions" for some computations of the fundamental volume of this tiling, but due to my loose knowledge of French, I cannot understand this text.

If you can tell me about an English translation of this text or if you can give me a reference that provides the same material in English, I would be very thankful.

Answer 1

There might be an English translation of Vigneras book. If not, at least there is Arithmetic of Hyperbolic 3-manifolds by Colin McLachlan and Alan Reid. That book has information about quaternion algebras.

I'll look into discussion of the fundamental volume of the tiling over $\mathbb{H}^3$.

In Chapter 11 they discuss Tamagawa measure. If $A$ is a quaternion algebra, and $\mathcal{O} \subseteq A $ is a maximal order then we've got a formula for the volume:

$$\text{Vol}\big( SL(2,\mathbb{C}) / \rho (\mathcal{O} ) \big)= \frac{|\Delta_k|^{3/2}\zeta_k(2)\prod_{\mathcal{P}\big|N(A)}\big( N(\mathcal{P})-1\big)} {(4\pi^2)^{[k:\mathbb{Q}]-2}}$$

This volume is expressed in terms of invariants of the quaternion algebra, which are in the textbook. For example $k/\mathbb{Q}$ is the number field which was used to define $A$, and $\zeta_k(2)$ is the Zeta function. There's another formula for volumes of $SL(2,\mathbb{R}) $ quotients as well.