# 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... 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 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) 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.
– user69015
Jun 17 '20 at 21:42
• Thanks all, appreciate it. 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 $$E_2$$).