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!



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


