Characteristics of an isogeny between super-singular elliptic curves

Question

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?

Answer 1

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 $E_2$.

As stated in Theorem 4.8 in page 71 of The Arithmetic of Elliptic Curves by J.H. Silverman:

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 $E_2$).

The proof involves the divisor class groups of the curves, using the fact that the isogeny induces a group homomorphism between them.