Timeline for Simple Tamagawa number calculations
Current License: CC BY-SA 3.0
14 events
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| Nov 5, 2012 at 3:49 | comment | added | Chandan Singh Dalawat | @Jonah: "You're reiterating what I wrote in my original question!" Not quite. The addendum was a way of pointing out that in the decomposition of $\mathrm{SL}_2(\mathbf{A})/\mathrm{SL}_2(\mathbf{Q})$ in your original question, the factor at $p$ should have been $\mathrm{SL}_2(\mathbf{Z}_p)$ and not $\mathrm{SL}_2(\mathbf{Q}_p)/\mathrm{SL}_2(\mathbf{Z}_p)$. | |
| Nov 3, 2012 at 17:26 | comment | added | GH from MO | @Chandan: I sent an email to the address I found in arxiv.org/abs/0711.3879 | |
| Nov 3, 2012 at 17:14 | comment | added | Chandan Singh Dalawat | @GH: All your comments and answers are very pertinent. Is it fair that only Jonah knows who you are ? | |
| Nov 3, 2012 at 16:57 | comment | added | Jonah Sinick | @ Chandan - I was thinking that the computation at the Archimedian component gives $\frac{{\pi}^2}{6}$ rather than $\zeta{(2)}$, so that the method that you suggested assumes the conclusion. I'll have to think more about these things in light of GH's comment elsewhere about the calculation of the volume as $\zeta{(2)}$ directly using Eisenstein series. | |
| Nov 3, 2012 at 16:49 | comment | added | GH from MO | @Chandan: The OP would like to explain Euler's formula by Weil's conjecture in a special case, together with a proof of that special case. If you calculate the Tamagawa number by local calculations, and proceed to explain Euler's formula that way, then all you do is calculating the the volume of SL(2,R)/SL(2,Z) in two ways and hide it in a fancy adelic setttings. The OP would like to use the adeles in a more genuine way. That can be done by adelic Eisenstein series, as done by Langlands in general (cf. my other comments here and also the addendum in my response). | |
| Nov 3, 2012 at 16:49 | comment | added | Jonah Sinick | @ GH - perhaps you're right :) Because of my background in hyperbolic geometry, that's the way that seems most natural to me. These things can be in the eye of the beholder. Anyway, note that my method doesn't require the calculation of a nontrivial integral (via classical methods from calculus), even if it involves several steps. | |
| Nov 3, 2012 at 16:40 | comment | added | Chandan Singh Dalawat | @Jonah: In general, you should consider yourself fortunate whenever you can resolve a global question into local questions at each place. The first reflex upon encountering a global problem should be to solve it locally everywhere. I don't see what anyone can have against this reflex. | |
| Nov 3, 2012 at 16:19 | comment | added | GH from MO | @Jonah: That sounds like a very complicated way to calculate the volume of SL(2,R)/SL(2,Z)! There are simpler ways as in the mentioned books by Platonov-Rapinchuk or Goldfeld. I can imagine that via adelic Eisenstein series one can prove directly that the Tamagawa number is 1. | |
| Nov 3, 2012 at 15:55 | comment | added | Jonah Sinick | Regarding your addendum: (1) You're reiterating what I wrote in my original question! I was asking for a proof that the Tamagawa number is 1 that doesn't rely on resolution into components. (2) the volume of SL(2,R)/SL(2,Z) can be computed by regarding: PSL( 2, R) is a fiber bundle over the upper half plane with fibers that are circles, computing the orbifold Euler characteristic of the modular domain in the upper half plane, and applying Gauss-Bonnet to its 24 fold genus 2 constant negative curvature cover. | |
| Nov 3, 2012 at 15:09 | history | edited | Chandan Singh Dalawat | CC BY-SA 3.0 |
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| Nov 3, 2012 at 12:48 | comment | added | user9072 | The book of Colmez seems a good choice for learning mathematical French, too (besides it being a fantastic book!). It starts with a 200pages chapter reviewing a large part of the standard math notions. The chapter is even called Vocabulaire Mathématique (Mathematical vocabulary). Here is a table of content editions.polytechnique.fr/files/pdf/TDMD_1587_9.pdf | |
| Nov 3, 2012 at 12:31 | history | edited | Chandan Singh Dalawat | CC BY-SA 3.0 |
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| Nov 3, 2012 at 7:14 | comment | added | Jonah Sinick | Thanks! This might give me motivation to learn enough French to read the text that you mention. | |
| Nov 3, 2012 at 6:54 | history | answered | Chandan Singh Dalawat | CC BY-SA 3.0 |