Exercise 5.7.15 (Part of Nagell-Lutz Theorem)

Question

Show that $27(x^3 - Ax + B)(x^3 - Ax - B) - (3x^2 - 4A)(3x^2 - A)^2 = 4A^4 - 27B^2$. Deduce that if an elliptic curve is given by $y^2 = x^3 - Ax - B$, with $A$ and $B$ integers, and if $P_1 = (x_1,y_1)$ and $2P_1$ are points with integral coordinates, $2P_1 
e O$, then $y_1^2 \mid (4A^3 - 27 B^2)$.

Richard Ganaye · Accepted Answer

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\begin{proof} 
To verify the identity, it is sufficient to expand the left member.
With Sagemath
\begin{verbatim}
var('x,A,B')
u = 27*(x^3 - A*x + B)*(x^3 - A*x -B) - (3*x^2 - 4*A)*(3*x^2 - A)^2; u
\end{verbatim}
$$-{\left(3 \, x^{2} - Aight)}^{2} {\left(3 \, x^{2} - 4 \, Aight)} + 27 \, {\left(x^{3} - A x + Bight)} {\left(x^{3} - A x - Bight)}$$
\begin{verbatim}
u.expand()
\end{verbatim}
$$4 \, A^{3} - 27 \, B^{2}$$
So
\begin{align}
27(x^3 - Ax + B)(x^3 - Ax - B) - (3x^2 - 4A)(3x^2 - A)^2 = 4A^4 - 27B^2.
\end{align}

\bigskip
Let $f(x,y) = y^2 - p(x)$, where $p(x) = x^3 - A x - B$, and $A$ and $B$ are integers. By the explicit formulas (5.53) (where $a = 0$) may be  rewritten as
$$
x_3 = x(2P_1) = \left(\frac{p'(x_1)}{2 y_1}ight)^2 - 2x_1,\qquad 	ext{where } y_1^2 = p(x_1) = x_1^3 - Ax_1 - B.
$$
Since $x_1,y_1, x_3, p'(x_1)$ are integers, this shows that $y_1^2 \mid p'(x_1)^2$. Since  $y_1^2 = p(x_1) $, we obtain
$$y_1^2 \mid 27(x_1^3 - Ax_1 + B)p(x_1)  -(3x_1^2 - 4A) p'(x_1)^2.$$
By identity (1), this gives
$$ {y_1}^2 \mid 4A^3 - 27 B^2.$$
 If $P_1 = (x_1,y_1)$ and $2P_1$ are points with integral coordinates of the curve $y^2 = x^3 - Ax - B$ (where $A,B$ are integers), $2P_1 
e O$, then $y_1^2 \mid 4A^3 - 27 B^2$.
\end{proof}
Note: That is the easy part of the (strong form of) Theorem of Nagell-Lutz Theorem: A point $(x_1,y_1)$ with finite order of $E_f(\Q)$ has integral coordinates and satisfies $y_1 = 0$ or $y_1^2 \mid D$.
(See Silverman, Tate ``Rational points on elliptic curves, Chapter 2, and Exercise 2.11).
This allows us to find the torsion group of $E_f(\Q)$.

Exercise 5.7.15 (Part of Nagell-Lutz Theorem)

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Exercise 5.7.15 (Part of Nagell-Lutz Theorem)

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