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Now I'm trying the section 6 (and 3.20) of chapter IV of Deligne-Rapoport's "Les schemas de module de courbes elliptiques".
I can't understand what $e_n$ (of 6.5.(d)) is. It seems to be the Weil pairing. So my question is:

Let $n$ a natural number, $S$ a scheme on which $n$ is invertible, and $C/S$ a generalized elliptic curve, whose each geometric fibre is smooth or the Neron $m$-gon for $m | n$. Then how can I define the Weil pairing on $C^\text{sm}[n]$?

Here is what I have tried.

First, if $C/S$ is smooth, then trivially we have the Weil pairing. And so it defines $\mathscr{M}_n^\circ[1/n] \to \mathbb{Z}[\zeta_n]$. ($\mathscr{M}_n$ is the stack classifying the generalized elliptic curves with the level $n$-structures. And $\mathscr{M}_n^\circ$ is the smooth part. And $\zeta_n$ is the primitive $n$-th root of unity.)
The authors say that by the definition of normailziation, this induces $\mathscr{M}_n \to \mathbb{Z}[\zeta_n]$. (I can't understand it, because $\mathscr{M}_n$ is the normalization of $\mathscr{M}_n^\circ[1/n] \to \mathscr{M}_1$.)

And even if we get $\mathscr{M}_n \to \mathbb{Z}[\zeta_n]$, this defines the Weil pairing only for generalized elliptic curves whose each fibre is smooth or $n$-gon.

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  • $\begingroup$ $e_n$ is explained in IV.3.21, isn't it? Moreover, at least if $n>5$, ${\mathcal M}_n[1/n]$ is the normalization of ${\mathcal M}_n^0[1/n]$ (and this is what you need, since $n$ is invertible in $S$ by hypothesis in 6.5). $\endgroup$
    – Nulhomologous
    Commented Sep 12, 2020 at 10:32
  • $\begingroup$ @Nulhomologous In 3.21, they define the Weil pairing only for $\mathscr{M}_m$, for $n|m$. I want to define it for a generalized elliptic curves whose fibres are smooth or $m$-gon, for various $m|n$. $\endgroup$
    – k.j.
    Commented Sep 13, 2020 at 8:51
  • $\begingroup$ But they show this parings for various $m\mid n$ are compatible with each other with respect to the natural morphisms, isn't it? $\endgroup$
    – Nulhomologous
    Commented Sep 13, 2020 at 10:10
  • $\begingroup$ @Nulhomologous Where is it? I can't find it. $\endgroup$
    – k.j.
    Commented Sep 14, 2020 at 13:25

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