Timeline for Computing Tamagawa number of torus in Quaternion algebra
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 2, 2018 at 12:35 | comment | added | Kimball | @DesideriusSeverus IIRC, Maclachlan and Reid do not do Tamagawa numbers, and I think Vigneras does Tamagawa numbers of norm 1 subgroups of quaternion algebras and quadratic fields, but not the quotient by the base field which will give you the factor of 2. | |
| Feb 2, 2018 at 8:47 | vote | accept | user113771 | ||
| Feb 1, 2018 at 16:51 | answer | added | Marty | timeline score: 6 | |
| Feb 1, 2018 at 8:46 | comment | added | user113771 | @Marty I also found Tam=Pic/Sha. But how to compute Pic or Sha in this setting? | |
| Feb 1, 2018 at 8:45 | comment | added | user113771 | @JohnVoight I need the quaternion algebra as I am interested in the centralizer $G_\gamma$. This can be larger than $\mathbb{Q}[\gamma]$. By generic I mean elements unlike $\gamma=1$ for which the centralizer is the whole Quaternion algebra. | |
| Feb 1, 2018 at 8:45 | comment | added | user113771 | @DesideriusSeverus Thanks, I will have a close look at Vignéras (or rather the translation). | |
| Feb 1, 2018 at 4:53 | comment | added | Marty | Tam = Pic / Sha, src = Ono. :) | |
| Jan 31, 2018 at 20:44 | comment | added | John Voight | What does "generic" mean? Your element $\gamma$ is an element of the quaternion algebra (up to rescaling), and so if $\gamma \not \in \mathbb{Q}$, then $\mathbb{Q}[\gamma] = K$ is an imaginary quadratic field. What is the role of the quaternion algebra here? Every imaginary quadratic field with the property that $2$ is not split in $K$ arises this way, so your question is about $K^\times/\mathbb{Q}^\times$ for these fields? My basic understanding is that the Tamagawa number is given by a class number, is that what you're after? | |
| Jan 31, 2018 at 14:14 | comment | added | Desiderius Severus | You should look at Vignéras, Arithmétique des Algèbres de Quaternions. A lot of computations are done there. I believe there is a translated version available online. A good part of this book can be found in Maclachlan and Reid, The Arithmetic of Hyperbolic 3-manifolds | |
| Jan 31, 2018 at 14:13 | review | First posts | |||
| Jan 31, 2018 at 14:33 | |||||
| Jan 31, 2018 at 14:10 | history | asked | user113771 | CC BY-SA 3.0 |