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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 $E'$ ?

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up vote 6 down vote accepted

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.

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I hope you are talking about something called the "Isogenies of Elliptic curves", I am sure that every Isogeny $\phi$ is a homomorphism, assuming that if $\phi : E \mapsto E'$ is faithful in carrying the identity element ( (or) point at infinity ) $O_E$ of $E$ to $O_{E'}$ of $E'$ , i.e. $\phi(O_E)=E'$ , so then what you said is right .

Thank you.

All the best.

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I also advice you to read the Silverman's text about elliptic curves for further details or write to me if you want to hear further – Iyengar Jan 4 '12 at 17:54

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