Timeline for Computation of low weight Siegel modular forms
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| S Feb 16, 2015 at 15:00 | history | suggested | user21574 | CC BY-SA 3.0 |
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| Feb 16, 2015 at 14:50 | review | Suggested edits | |||
| S Feb 16, 2015 at 15:00 | |||||
| Apr 30, 2010 at 6:29 | comment | added | Kevin Buzzard | @David: in which case my answer to your aside is then simply "yes". | |
| Apr 30, 2010 at 0:48 | comment | added | David Hansen | @Kevin: I am aware of the low-dimensional coincidences SO(2,1)=PGL(2) and Spin(2,3)=Sp(4). I suppose n > 2 was implicit in my query. :) | |
| Apr 30, 2010 at 0:02 | comment | added | Kevin Buzzard | @David: there is a low-degree coincidence: the dual group of GSp_4 is GSp_4 (think about the Dynkin diagram of Sp_4). I agree that the pattern should not continue for Sp_6 (think about the Dynkin diagram again). | |
| Apr 29, 2010 at 20:52 | comment | added | David Hansen | Aside: Shouldn't the automorphic representation associated to an abelian n-fold live on SO(2n+1), rather than a symplectic group? (Look at the L-groups) | |
| Apr 29, 2010 at 20:14 | comment | added | Kevin Buzzard | Thanks for the reference FC. What you say about Sp_6 is terrifying! | |
| Apr 29, 2010 at 19:16 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
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| Apr 29, 2010 at 19:15 | comment | added | Kevin Buzzard | @David: I think that trying to describe the moduli space is not what I am after. I am after an algorithm to compute traces of Hecke operators on spaces of cusp forms that takes a level and a prime (or some slightly more precise data; a diagonal matrix with powers of p on the diagonal) as an input and gives a number as an output and doesn't care less about whether a certain moduli space is general type or not. The same way as modular symbols would do the job for classical modular forms. But from FC's comment it seems that the trace formula won't do it either :-/ | |
| Apr 29, 2010 at 19:13 | comment | added | Kevin Buzzard | @FC: I am pretty sure one can't use the trace formula to compute forms of weight 1 but I have so little understanding of the trace formula that I don't know why this is the case. Seems to me that you're saying that the same obstruction stops me computing low weight Siegel forms :-( | |
| Apr 29, 2010 at 17:06 | comment | added | David Lehavi | @Kevin: Curves of genera up 6 have a very pleasant description, and you can comfortably (depends on your standard of comfort of course) live with the description of curves up to genus 15; I'd buy a curve of genus up to 15 as a moduli space any day. I have yet to see a general type three-fold with a nice description (unless you cooked it for this purpose). Disclaimer: I don't know the method involved, you may well be right, it's just my vague intuition speaking here. | |
| Apr 29, 2010 at 14:52 | comment | added | Kevin Buzzard | I don't buy this. Isn't that like saying "wouldn't computational efforts with elliptic curves of conductor 100 or more be hard because the moduli space has genus at least 2", isn't it? The point is that you don't compute the moduli space---that's the last thing you want to do! You use the trace formula applied to a carefully-chosen function which will give you essentially the trace of Frobenius on the cohomology in terms of some much more algebraic/combinatorial/number-theoretic data and compute that instead. Well, that's my suggestion, but a lot of thought needs to go in to making it work. | |
| Apr 29, 2010 at 13:58 | comment | added | David Lehavi | Wouldn't any computational efforts with N > 3 be very difficult simply because the moduli space is general type ? | |
| Apr 29, 2010 at 6:46 | history | asked | Kevin Buzzard | CC BY-SA 2.5 |