Timeline for Representations attached to p-adic modular forms
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 11, 2012 at 1:48 | comment | added | Joël | characteristic polynomial is needed. The correct notion was developed by Chenevier, under the name "determinants", and it works fine in any characteristic. The article is on his web page. | |
| Dec 11, 2012 at 1:47 | comment | added | Joël | Then come's Rouquier (and Nyssen). Rouqier's notion of pseudo-character is essentially the same as Taylor's. Ho offers a nice, self-contained treatment, and states and prove Taylor's theorem for an alg. closed field of any characteristic. Unfortunately, there is an unfixable mistake if the characteristic is less or equal than the dimension. So his theorem is proved only in char greater than dimension. The problem is in the notion of Rouquier's or Taylor's pseudocharacter/pseudorepresentation: that notion focuses on the trace of a representation, while in low characteristic, the full... | |
| Dec 11, 2012 at 1:43 | comment | added | Joël | @Filipo. I apologize, I haven't seen your comment earlier. I don't know if Taylor and Wiles definition are equivalent in dim 2 and char 2. What I know is that both notions are not the good one. To summarize, Wiles' notion os the older, and is very peculiar to the situation he was studying. Taylor's notion makes senses in any dimension, and any characteristic, but the main theorem (that over an alg. closed field any character comes from a true representation) is proved only by him in char 0. (Note that this is enough for the argument of my answer to the PO's question)... | |
| Oct 13, 2012 at 3:00 | comment | added | Filippo Alberto Edoardo | @Joël: Do you have an explicit reference for the equivalence of Taylor's and Wiles' definition of pseudo-character in dimension $2$ (may be assuming $2$ is invertible in the ring)? I tried, but I was submerged by computations which I was unable to bring to an end....Thanks! | |
| Sep 29, 2012 at 1:47 | history | edited | Joël | CC BY-SA 3.0 |
Improved the latex.
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| Sep 28, 2012 at 20:00 | comment | added | Joël | Yes, I should add that the argument outlined above is essentially present (in a different context) in Richard Taylor's thesis. I am not exactly sure if it is due to Taylor or Wiles (who invented the pseudo-characters in dimension 2), or Mazur-Wiles, but assuredly it comes from this group of people in the late 80's. | |
| Sep 28, 2012 at 19:47 | vote | accept | Arijit | ||
| Sep 28, 2012 at 19:33 | comment | added | Arijit | Ahh this is wonderful. Thank you. As far as I can remember from Gouvea's book(quoting it from memory) he needed the condition of absolute irreducibility as he was using Mazur's result on the existence of universal deformation ring. But this completely circumvent that problem. Thanks again for your answer. | |
| Sep 28, 2012 at 19:06 | history | edited | Kevin Ventullo | CC BY-SA 3.0 |
Fixed some typos
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| Sep 28, 2012 at 18:19 | history | answered | Joël | CC BY-SA 3.0 |