Exercise 5.7.14 ($2P$ has integers coordinates)

Question

Let $P_1 = (x_1, y_1)$ be a point with integral coordinates on the elliptic curve $y^2 = x^3 + ax^2 + bx+c$, where $a,b,c$ are integers. Show that if $2P_1$ also has integral coordinates then $(2y_1) \mid (3x_1^2 + 2ax_1 + b)$.

Richard Ganaye · Accepted Answer

\begin{proof} We suppose that $2P_1 = (x_3, y_3)$ has integral coordinates. By the explicit formulas (5.53),
$$x_3 = \left(\frac{3x_1^2 + 2a x_1 + b}{2y_1 }\right)^2 -a - 2x_1.$$
Since $x_1, x_3, y_1, a, b$ are integers, this shows that
$$ (2y_1)^2 \mid (3x_1^2 + 2a x_1 + b)^2.$$
By the unique factorization Theorem, this implies
$$2y_1 \mid 3x_1^2 + 2a x_1 + b.$$

\end{proof}

Exercise 5.7.14 ($2P$ has integers coordinates)

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Exercise 5.7.14 ($2P$ has integers coordinates)

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