## Cubic curves

Dear Professor Kim,

Thank you for your reply. I have another question, this time regarding the group law on a cubic, . I am happy with showing that the points of 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 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 does it correspond to in affine 2-space?

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Reply:

IF is the curve in , then it meets the line in the point as explained in the book. So we then express as a union

where We then note that

by the map

Then goes to the curve in with equation

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

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

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