I am scuffling with the proof of two.21 of Saito’s “Fermat’s Final Theorem”.

Let $omega$ be a primitive third root of unity, $X(3) = mathbb{P}^1_{mathbb{Q}(omega)}$, and $E = { X^3 + Y^3 + Z^3 – Three mu XYZ } subseteq mathbb{P}^2_{X(3)}.$

(the place $mu$ is an inhomogeneous coordinate of $X(3)$.)

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

With a view to present that $X(3)$ is the advantageous moduli scheme (over $mathbb{Q}$) of full stage Three construction, I wish to present that $E$ has a construction of a generalized elliptic curve with the $0$-section $[0 : 1 : -1]$, such that $P, Q$ is a foundation of the $3$-torsion factors.

Here’s what I attempted:

Let $Y(3) = X(3) – { 1, omega, omega^2, infty}$.

Then $E$ is clean 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)},

E_omega = { (X + Y + omega Z)(X + omega Y + Z)(X + omega^2 Y + omega^2 Z)},

E_{omega^2} = { (X + Y + omega^2 Z)(X + omega^2 Y + Z)(X + omega Y + omega Z)}.$$

These fibres are Neron 3-gon, so we are able to outline generalized elliptic curve buildings on them, such that the fibres of $P, Q$ are bases of their 3-torsion factors.

How can I outline the generalized construction on $E$ globally?

(It appears for me that since $E$ is a plain cubic curve, we are able to outline $E^textual content{sm} instances E to E$ utilizing Bézout’s theorem.)

Thanks very a lot!