# Characteristics of an isogeny between super-singular elliptic curves

I believe I've read this before, but I can't find it despite hours of searching on Google.

I've know the common definition of isogeny in elliptic curves, as $$\phi:E_1 \rightarrow E_2$$ a nonconstant morphism of curves satisfying $$\phi(0)= 0$$, however somewhere else I read a really lovely intuitive explanation that basically stated that if a curve $$E_1$$ is isogenic with curve $$E_2$$, then, not only will points, such as $$p_1$$, $$q_1$$ map to $$p_2$$, $$q_2$$ but moreover, the mapping of the point $$E_1(p_1+q_1)$$ to $$E_2$$, will result in the same point on $$E_2$$ as when the points $$E_1(p_1)$$ & $$E_1(q_1)$$ are first mapped to $$E_2$$ and then added on the $$E_2$$ curve.

I hope this makes sense, am I confusing this or does anyone have a resource that shows this detail?

• Actually, preserving the group operation is what "morphism" means; that $\phi(p_1+q_1) = \phi(p_1) + \phi(q_1)$. Just a random mapping that isn't constrained to preserve the group properties wouldn't be very interesting... – poncho Jun 17 '20 at 20:39
• Ok so my take is correct? Do you have any link to docs that show this characteristic? I can’t find it anywhere. Thanks @poncho – Woodstock Jun 17 '20 at 20:47
• Other than the Wikipedia article on "morphism", well, no, I don't have one available. It's one of those ideas that's so "in the water" that it's usually just assumed everyone knows (and so there's little point in spelling it out) – poncho Jun 17 '20 at 21:12
• See theorem 4.8 on page 71 of The Arithmetic of Elliptic Curves by Silverman. It proves that an isogeny between elliptic curves is a group homorphism. – corpsfini Jun 17 '20 at 21:42
• Thanks all, appreciate it. – Woodstock Jun 17 '20 at 21:43

Given two elliptic curves $$E_1$$ and $$E_2$$, an isogeny from $$E_1$$ to $$E_2$$ is an algebraic map $$\phi$$ (it is basically defined from rational functions) that sends the neutral element from $$E_1$$ to the neutral element of $$E_2$$.
Let $$\phi : E_1 \longrightarrow E_2$$ be an isogeny. Then $$\phi(P+Q) = \phi(P) + \phi(Q)\quad \text{for all } P, Q \in E_1.$$
In other words, the isogeny is a group homomorphism (the group law of the curve $$E_1$$ is preserved after the transfer to the curve $$E_2$$).