where is the difference between an elliptic curve in affin and in projective representation? I know that an projective elliptic curve can create an abelian group. Does the affine elliptic curve does the same? Thank you very much!
2026-04-07 07:46:10.1775547970
Difference between affine and projective elliptic curve
1.2k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
By definition, an elliptic curve is a smooth projective curve of genus 1 with a rational point. So every time you read something like "Let $y^2=x^3+Ax+B$ be an elliptic curve..." this is a slight abuse of notation: such an equation is only the affine part of an elliptic curve. You can recover the equation of the corresponding projective curve simply by homogeneizing the equation with respect to a new variable $z$. In this way, you are adding only a point to the curve, on the line $z=0$, that will play the role of the identity in the group structure, and so it is of the utmost importance. When you read that an elliptic curve admits an abelian group structure, this means that the set of rational points on the projective equation of your curve admits a structure of abelian group (compatible with the structure of algebraic curve, of course). I believe, but I am not completely sure about this, that there is no way to make the set of affine points on an elliptic curve into an abelian group in a compatible way with the algebraic nature of the curve.