An explicit description of $X(3)$ and its universal generalized elliptic curve

Question

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.:

1. 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...

2. 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!

Answer 1

I have understood.

1. $E$ is normal connected scheme.

First, since $X$ is connected and $E \to X$ is proper flat of finitely presentation, $E$ is connected.
Next, since $E^\text{sm} \to X$ is smooth and since $X$ is regular, $E^\text{sm}$ is regular. Now since $E \to X$ is local complete intersection, in particular Cohen-Macaulay, and is 2-dimensional, $E$ is normal.

2. The rational map $\varphi : E^\text{sm} \to \operatorname{Isom}(E, E), P \mapsto (Q \mapsto P+Q)$, a priori defined on $E \times_X Y$, is defined everywhere.

By (1.3) of Artin's Neron Model (in Cornel-Silverman's Arithmetic Geometry), it suffices to see that this $\varphi$ is defined at $x \in E^\text{sm}$, which is a generic point of $E^\text{sm}_s$. (Where $s$ is the image of $x$ under $E \to X$. Now if $s$ is the generic point, then since $\varphi$ is defined at $x$, it suffices to check this for closed $s$.)
Since $E^\text{sm} \to X$ is flat, such a point is exactly a point which is codimension 1 in $E^\text{sm}$. Thus it suffices to see that the morphism $\operatorname{Spec}k(E^\text{sm}) \to E^\text{sm} \to \operatorname{Isom}$ is extended to $\operatorname{Spec}\mathscr{O}_{E^\text{sm}, x} \to \operatorname{Isom}$. (Where $k(E)$ is the function field of the scheme $E$.)

So we show:

Let $R$ be a DVR with the prime element $\pi$, the residue field $k$, and the fractional field $K$ of charactericstic $\neq 3$. Let $E$ be a curve over $R$, defined by $$X^3 + Y^3 + Z^3 - 3 \mu XYZ = 0$$ or $$\mu (X^3 + Y^3 + Z^3) - 3 XYZ = 0.$$ ($\mu \in R$). Assume that $E$ is smooth over $K$ and is singluar over $k$.
Let $f : E_K \to E_K$ be an isomorphism over $K$. Then this $f$ extends to a isomoprhism $g : E \to E$ over $R$.

First, by (4.1.16) of Liu's Algebraic Geometry and Arithmetic Curves, the morphism $f : E_K \to E_K$ extends to a rational map $g : E \to E$, defined at every codimension 1 point.
On the other hand, write $\mu = \epsilon \pi^n$. Since $E_k$ is singular, $n \ge 1$. Now blowing-up $E$ at every singular point $\lfloor n/2 \rfloor$-times, we get the regular model $\mathfrak{E}$ over $R$, of $E_K$. Seeing its special fibres, we can see that $\mathfrak{E}$ is minimal. Thus $f : E_K \to E_K$ extends to the isomorphism $\mathfrak{E} \to \mathfrak{E}$. Since $\require{AMScd}$ \begin{CD} \mathfrak{E} @>{h}>> \mathfrak{E}\\ @V{p}VV @V{p}VV\\ E @>{g}>> E \end{CD}

is commutative, this $\mathfrak{E} \to \mathfrak{E}$ induces $g : E \to E$, defined everywhere. (First, since $g$ is defined on a dense open, we can show that $h(\text{a contractible line}) = (\text{a contractible line})$. On the other hand, since $p: \mathfrak{E} \to E$ is a closed map, $E$ is some quotient of $\mathfrak{E}$ as a topological space. Thus $h$ induces $g$ as topological spaces. Finally, since $p_*\mathscr{O}_\mathfrak{E} = \mathscr{O}_E$, $h$ induces a map of schemes $g$.)
(Or, considering affine neighbourhood, we can reduce the situation into an easy ring lemma: Let $A$ be a normal noetherian local ring of dimension $2$ and let $f : \operatorname{Spec}A - \{ \mathfrak{m}\} \to \operatorname{Spec}A - \{ \mathfrak{m}\}$ be a morphism of schemes. Then this $f$ is defined everywhere, i.e., defines the ring endomorphism on $A$. Since now $A = \cap A_\mathfrak{p}$, where $\mathfrak{p}$ runs through all prime ideals of $A$ of heght $1$, this is almost trivial.)

3. The morphism $\varphi : E^\text{sm} \to \operatorname{Isom}(E, E)$ defines a group structure on $E^\text{sm}$ and defines a group action of $E^\text{sm}$ on $E$.

Since the "group axiom diagrams" and "the group action axiom diagram" commute over $Y$, and since $E^\text{sm} \times E \times E$ is reduced and since $E$ is separated, these diagrams are commutative.

4. This is a generalized elliptic curve.

By II.1.15 of Deligne-Rapoport, this is trivial.

Finally,

5. $P$ and $Q$ defines the full level $3$ structure.

Over $Y$, by Bezout. ($E \cap \{ \mu \omega^2 X + Y + \omega Z = 0 \} = 3P$ and $E \cap \{ \mu \omega X + Y + \omega^2 Z = 0 \} = 3Q$ as Cartier divisors.)
Over $X$, checking at each fibre, $(\mathbb{Z}/3)^2 \to E^\text{sm}[3]$ is an isomorphism fibre-by-fibre. Thus is an isomorphism globally.