I'm struggling with the proof of 2.21 of Saito's "Fermat's Last Theorem".

Let $\omega$ be a primitive 3rd root of unity, $X(3) = \mathbb{P}^1_{\mathbb{Q}(\omega)}$, and $E = \{ X^3 + Y^3 + Z^3 - 3 \mu XYZ \} \subseteq \mathbb{P}^2_{X(3)}.$ (where $\mu$ is an inhomogeneous coordinate of $X(3)$.)

Let $O = [ 0:1:-1], P= [0:\omega:-1], Q = [1:0:-1].$

In order to show that $X(3)$ is the fine moduli scheme (over $\mathbb{Q}$) of full level 3 structure, I want to show that $E$ has a structure of a generalized elliptic curve with the $0$-section $[0 : 1 : -1]$, such that $P, Q$ is a basis of the $3$-torsion points.

Here is what I tried:

Let $Y(3) = X(3) - \{ 1, \omega, \omega^2, \infty\}$.
Then $E$ is smooth over $Y$.

And for $\mu \in X - Y$, the fibre of $E$ at $\mu$ is

$$E_\infty = \{ XYZ = 0 \}, \\ E_1 = \{ (X + Y + Z)(X + \omega Y + \omega^2Z)(X + \omega^2 Y + \omega Z) = 0 \}, \\ E_\omega = \{ (X + Y + \omega Z)(X + \omega Y + Z)(X + \omega^2 Y + \omega^2 Z) = 0 \}, \\ E_{\omega^2} = \{ (X + Y + \omega^2 Z)(X + \omega^2 Y + Z)(X + \omega Y + \omega Z) = 0 \}.$$

These fibres are Neron 3-gon, so we can define generalized elliptic curve structures on them, such that the fibres of $P, Q$ are bases of their 3-torsion points.

How can I define the generalized structure on $E$ globally?

**Edit**

Seeing a comment of Will Sawin, I have 2 idea of defining a generalized elliptic curve law.:

Using that $E$ is a plain cubic curve, define $E^\text{sm} \times E \to E$ by Bézout's theorem.

First writing down the explicit addition law for $E \times_{X(3)} Y(3)$, then define $E \times E \to E$ by that explicit formula. And check the group action for $E^\text{sm}$ and the group action axiom for $E^\text{sm} \times E \to E$.

But the addition law is too complicated to write down explicitly...Extend the morphism $E \times Y(3) \to \operatorname{Isom}(E, E) \times Y(3) : P \mapsto (Q \mapsto P + Q)$ to the morphism $\varphi : E^\text{sm} \to \operatorname{Isom}(E, E)$.

If for each closed point $s$ of $X(3)$ and for generic point $x$ of $E_s$ the morphism $\varphi$ is defined at $x$, then by a general theorem (see proposition 1.3 of Artin's Neron models), we have that $\varphi$ is defined everywhere. (Now since $E$ is in the projective space, the Hilbert scheme exits, hence so does the scheme of isomorphisms.)

To define $\varphi$ for a such point $x$, it sufficies to see that $\operatorname{Isom}$ is proper over $X(3)$. (for example, see 4.1.16 of Liu's Algebraic geometry and arithmetic curves.) But I think this is false, since for a Neron n-gon $C$ over a field, $\operatorname{Aut}(C)$ is not proper. (see II 1.8 of Deligne-Rapoport)

Thank you very much!

10more comments