I was wondering whether an admissible change of variables of an elliptic curve given by
a Weierstrass equation respects the group law.
Let $E$ be defined over a field $K$ given by the equation
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$
If we substitute
$$(X,Y)\mapsto (u^2X+r,u^3Y+su^2X+t),$$
where $u\in K^*$ and $r,s,t \in K$, we get an elliptic curve $E'$. Denote the above transformation by $\phi$. Taking affine points $P_1,P_2 \in E$, is it true that
$\phi(P_1+P_2)=\phi(P_1)+\phi(P_2)$ on $E'$ ?
[Math] admissible change of variables on elliptic curve
elliptic-curves
Best Answer
Yes. Your transformation $\phi$ is an affine linear transformation, so it takes straight lines to straight lines, and therefore three points $P,Q,R$ are collinear if and only if $\phi(P),\phi(Q),\phi(R)$ are collinear. Since the group law is defined by the fact that $P+Q+R=0$ if and only if $P,Q,R$ are collinear, your result follows.