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 ?



Overconvergent modular forms and the level at $p$
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).


