3
$\begingroup$

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.

$\endgroup$
6
  • 1
    $\begingroup$ Did you try John Voight's book in progress math.dartmouth.edu/~jvoight/quat.html ? $\endgroup$ – Piotr Achinger Oct 4 '17 at 19:05
  • 5
    $\begingroup$ 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. $\endgroup$ – LSpice Oct 4 '17 at 20:12
  • $\begingroup$ @PiotrAchinger Thank you! It seems to be a useful reference. $\endgroup$ – Breakfastisready Oct 4 '17 at 22:14
  • $\begingroup$ @LSpice Thanks for the suggestion. I am in the pursuit of learning French and your suggestion certainly motivates me. $\endgroup$ – Breakfastisready Oct 4 '17 at 22:15
  • $\begingroup$ yumpu.com/en/document/view/7921447/… $\endgroup$ – Franz Lemmermeyer Oct 5 '17 at 4:41
2
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$ – john mangual Oct 4 '17 at 20:18
  • $\begingroup$ Thanks a lot for the information. That was certainly helpful. $\endgroup$ – Breakfastisready Oct 4 '17 at 21:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.