Timeline for Overconvergent modular forms and the level at $p$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2022 at 21:27 | history | bounty ended | babu_babu | ||
| Jun 20, 2022 at 21:27 | vote | accept | babu_babu | ||
| Jun 19, 2022 at 17:28 | comment | added | babu_babu | [cont. ] but Coleman seems to only ask that the image of $\mu_{p}$ and $\mu_{p^{m-1}}$ under the level structure at $p$ equals the canonical subgroup. Doesn't this mean that in reality the fibers are too big ? Shouldn't the correct definition (to make Chenevier's claim true) be that the image of $\mu_{p^m}$ is the canonical subgroup of order $p^m$ (i.e. that ``$\Phi^m(\alpha(\mu_{p^m})) = 0$'' in Coleman's notation) ? | |
| Jun 19, 2022 at 17:25 | comment | added | babu_babu | Also [sorry I deleted this from the original question to make it shorter but I figure I might try my luck and ask it again], regarding Coleman's conditions on the level structure, it seems to me that this description is the way to prove Chenevier's claim (at the top of p. 6 of the arXiv version of the J-L paper) that (for $v$ in the range where enough canonical subgroup exists) the $v$-overconvergent locus in $X_1(Np)$ is identified with the quotient of the $v$-overconvergent locus in $X_1(Np^m)$ by the diamonds in $(1 + p\mathbf{Z}_p)/(1 + p^m\mathbf{Z}_p)$. | |
| Jun 19, 2022 at 17:24 | comment | added | babu_babu | Thank you ! I have been very intimidted by Katz--Mazur for a long time, but maybe now it will be more realistic for me. | |
| Jun 19, 2022 at 17:18 | comment | added | David Loeffler | I suggest you start by reading Katz--Mazur. I'm not sure if the variation of the number of components with the overconvergence radius is a terribly natural question actually; the interest is in the limit as $v$ goes to 0. | |
| Jun 19, 2022 at 17:14 | comment | added | babu_babu | Great, thank you ! It is very nice to hear from the experts on $p$-adic automorphic forms on this site :) Do you know a reference that I could look at for this not entirely obvious fact, and in general for understanding better these connected components ? For example, is there a way to predict the number of connected components as a function of the overconvergent radius you choose ? | |
| Jun 19, 2022 at 17:08 | history | answered | David Loeffler | CC BY-SA 4.0 |