## Cubic curves

Dear Professor Kim,

Thank you for your reply. I have another question, this time regarding the group law on a cubic, $C$. I am happy with showing that the points of $C$ form a group. However I am a little unsure of the ‘simplified’ group law as described in the textbook. I believe that if we take $O=(0,1,0)$ as the identity element we can treat the cubic as an affine curve. However as this corresponds to the ‘line at infinity’ in projective 2-space, what point $(x,y)$ does it correspond to in affine 2-space?
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IF $C$ is the curve $Y^2Z=X^3+axZ^2+bZ^3,$ in $P^2$, then it meets the line $Z=0$ in the point $O=(0:1,0)$ as explained in the book. So we then express $C$ as a union

$C=\{O\}\cup C_0$

where $C_0=C\cap \{(X:Y:Z) | Z\neq 0\}.$ We then note that
$\{(X:Y:Z) | Z\neq 0\}\simeq A^2$ by the map

$(X:Y:Z)\mapsto (X/Z,Y/Z).$

Then $C_0$ goes to the curve in $A^2$ with equation

$y^2=x^3+ax+b.$

Of course the origin does not lie on this curve, because this affine curve corresponds exactly to the complement of the origin in $C$.

Does this answer your question? Let me know again if you’re still curious.