# English translation of M.-F. Vigneras “Arithmétique des algèbres de quaternions”

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.

• Did you try John Voight's book in progress math.dartmouth.edu/~jvoight/quat.html ? – Piotr Achinger Oct 4 '17 at 19:05
• In addition to any other answers, it is fair to say that you should probably learn to read mathematical French. (This is unavoidable for me as a representation theorist, but I imagine it's true in many disciplines.) It's way easier to learn the mathematical version than even the most basic conversational version of the language. – LSpice Oct 4 '17 at 20:12
• @PiotrAchinger Thank you! It seems to be a useful reference. – Breakfastisready Oct 4 '17 at 22:14
• @LSpice Thanks for the suggestion. I am in the pursuit of learning French and your suggestion certainly motivates me. – Breakfastisready Oct 4 '17 at 22:15
• yumpu.com/en/document/view/7921447/… – Franz Lemmermeyer Oct 5 '17 at 4:41

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.
• This is not my specialty. I am using this answer as learning opportunity. Also interested in the use of $\zeta_F (2)$ for number fields. – john mangual Oct 4 '17 at 20:18