Exercise 5.7.7 (Addition law on an elliptic curve)

Question

Let $A$ and $B$ be distinct points on an elliptic curve $\mathscr{C}_f(\mathbb{R})$, and suppose that the line trough $A$ and $B$ is tangent to $\mathscr{C}_f(\mathbb{R})$ at $B$. Show that $A + 2 B = OO$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Since the line trough $A$ and $B$ is tangent to $\mathscr{C}_f(\mathbb{R})$ at $B$, we obtain $BB = A$, thus $B + B = O(BB) = OA$.

Let $C = 2B = OA$. Then $AC = O$, thus
$$A +2B = A + C = O(AC) = OO.$$
This shows that
$$A + 2B = OO.$$
\end{proof}

Exercise 5.7.7 (Addition law on an elliptic curve)

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Exercise 5.7.7 (Addition law on an elliptic curve)

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