# Overconvergent modular forms and the level at $p$

I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.

The question concerns the relationship between various definitions of overconvergent modular forms in the standard papers of Coleman ("Banach spaces and families of modular forms"), Katz ("p-adic properties of ..."), and Chenevier ("une correspondence de Jacquet-Langlands p-adique" https://arxiv.org/abs/math/0301032), where section 3 contains a very brief but useful description of the basic setup).

In section B.2 of Coleman, if $$0 \leq v < p^{2-m}/(p+1)$$, the definition of the $$v$$-overconvergent locus in $$X_1(Np^m)$$ is a little bit complicated: you need to enforce the condition $$v(E_{p-1}) \leq v$$ as usual, but then you also add some conditions related to the level structure, as follows. For a given point $$x$$ representing the data $$(E, \alpha_N, \alpha_p)$$ [the level structure at $$p$$ being $$\alpha_p : \mu_{p^m} \to E[p^\infty]$$], in order to include $$x$$ in the $$v$$-overconvergent locus, Coleman asks that the image of $$\alpha_p|_{\mu_p}$$ is the canonical subgroup of order $$p$$, and that the image of $$\alpha_p|_{\mu_{p^{m-1}}}$$ is the canonical subgroup of order $$p^{m-1}$$ (at least this is my interpretation of what is going on in section B.2). On the other hand, in Chenevier, it is simply defined to be the connected component of $$\infty$$ in the locus where $$v(E_{p-1}) \leq v$$. Why are these definitions the same ? Is the point that $$X_1(Np^m)$$ itself might not be connected, or is the point that removing the too supersingular discs makes it disconnected ? Also, what is the relationship between all of this and the definition in Katz of "modular forms with growth condition" as rules defined on tuples of modular data ?

• If you don't get answers here in a few days, maybe try posting 1 question at a time on MathOverflow. Jun 17, 2022 at 21:26
• @Kimball : Dear Professor Martin: Sorry, I am not a Mathoverflow expert yet. Is it considered bad manners to post a question with multiple sub-parts like this one ? I did it this way because the questions are very closely linked, and an answer to one of them will probably go much of the way towards answering the others. I also don't want to flood the site with many similar questions, or be that person who posts another very similar question as soon as the previous one is answered --- in some sense this seemed to be the only way to explain the full context of my question. Jun 17, 2022 at 22:14
• For some reason I thought this post was on MSE (I must have gotten my tabs mxied up) and the main thing I was saying is MO is a better fit than MSE. A general guideline, though not a strict rule, is to ask just 1 focused question per post. This makes people more willing to read and answer. My suggestion is to try to refine your post (edit) to 1 or 2 briefer questions, and then if you need to ask a follow up question later you can. Jun 18, 2022 at 12:20
• @Kimball: Thank you for the advice ! I have made the question shorter, as suggested, and completely removed the fourth question (there's a good chance that an answer to what is here already will also answer it). Hopefully it will be more attractive now. Jun 18, 2022 at 12:42

The curve $$X_1(Np^n)$$ is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union of several components. What you want for a good theory of overconvergent forms is to pick out just one of these components. This can be done using conditions on the level structure (as in Coleman) or by taking the component containing the cusp $$\infty$$ (as in Chenevier).
It's fairly straightforward to check that the cusp $$\infty$$ does lie in the locus defined by the conditions you quote from Coleman, so the definitions do agree (at least if you believe that the locus cut out by Coleman's conditions is a connected component, which is not entirely obvious).
• 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 17:14
• 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 17:18
• 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 17:25
• [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 17:28