I am interested in understanding the computation of the number of cusps of $X_1(N)$, which is the modular curve that parameterizes the space of pairs of a generalized elliptic curves $E$ and a point $P$ in $E[N]$ of exact order $N$.
Inside it sits the modular curve $Y_1(N)$, which parameterizes pairs of elliptic curves with a point of exact order $N$.
A theorem of Deligne and Rapoport says that a generalized elliptic curve is either an elliptic curve or a Neron polygon, and the set of cusps of $X_1(N)$ should supposedly be the number of pairs $(C_m, P)$, where $C_m$ is a Neron polygon of length $m$, and $P$ is a point of exact order $N$.
The smooth locus of $C_m$, denoted $C_m^{sm}$, has a group structure which is given by $\mathbb{G}_m\times \mathbb{Z}/m\mathbb{Z}$, and a point of exact order $N$ should come from that locus.
For simplicity, let's assume $N$ is prime.
- What are the points of exact order $p$ on $C_m$?
It seems to me that a torsion element $(a,b)$ of a group $A\times B$, where $A$ and $B$ are groups, must consist of a pair such that $a\in A$ is torsion and $b\in B$ is torsion. In this case, if $a$ has order $l$ and $b$ has order $k$, then $(a,b)$ should have order $LCM(l,k)$. In that case, since $p$ is prime, an order $p$ element of $\mathbb{G}_m\times \mathbb{Z}/m\mathbb{Z}$ must be of the form $(\zeta_p,0)$, where $\zeta_p$ is a $p$'th root of unity, or of the form $(1,(m/p)*u)$, with $(u,p) = 1$, the second type of elements exists only when $p|m$, but should exist for every $m = kp$.
In the literature, it seems that the cusps of $X_1(p)$ only come from Neron polygons $C_1$ and $C_p$, but my argument shows that there should be cusps associated to $C_{kp}$ for ever $k\in\mathbb{N}$, where am I wrong?
Trying to count the number of cusps coming from $C_1$ according to my argument, shows that $C_1$ has $p-1$ cusps associated to it coming from elements of the form $(\zeta_p,0)$, however, in the literature, authors mod out by the automorphism group of $C_1$, which results in $(p-1)/2$ cusps. Why do we need to mod out by the automorphism group of $C_1$, when we do not mod out anything for an elliptic curve which is not a Neron polygon?
Trying to count the number of cusps coming from $C_p$, one seems to only be interested in cusps associated to elements of the form $(1,mu/p)$, where $(u,p) = 1$. There are $p-1$ such elements, but after we mod out and count orbits of the automorphism group we get $(p-1)/2$. My question is: what about the elements in $C_p$ of exact order $N$ of the first type, i.e. $(\zeta_p,0)$, why are these not counted as well?
Thanks in advance!