### Isogenies

#### Definition

[Was08, Chapter XII.1] Let $$E_1:y^2=x^3+a_1x+b_1$$ and $$E_2:y^2=x^3+a_2x+b_2$$ be elliptic curves over a field $$K$$. An isogeny from $$E_1$$ to $$E_2$$ is a nonconstant homorphism $$\alpha:E_1 \rightarrow E_2$$ that is given by rational functions.

This means $$\alpha(P+Q)=\alpha(P)+\alpha(Q)$$ for all $$P,Q \in E_1$$ and there exists rational functions $$P, Q$$ such that if $$\alpha(x_1, y_1)=(P(x_1, y_1),Q(x_1, y_1))$$.

In fact, it can be proved that we can write $$\alpha$$ in the form $$\alpha(x_1, y_1)=(p(x_1), y_1q(x_1))$$.

If $$p(x)=\dfrac{r(x)}{s(x)}$$ for polynomials $$r$$ and $$s$$ without common roots, define the degree of $$\alpha$$ to be $$Max(deg(r(x)),deg(s(x)))$$.

We say an isogeny is seperable if $$s(x)$$ have no repeated roots.

#### Example

Consider two curves $$E_1:y^2=x^3+x$$ and $$E_2:y^2=x^3-4x$$ over $$\mathbb{F}_{11}$$. Then the map $$\alpha: E_1 \rightarrow E_2$$ $$(x,y) \mapsto \left(\dfrac{x^2+1}{x},\dfrac{y(x^2-1)}{x}\right)$$ is an isogeny from $$E_1$$ to $$E_2$$.

#### Properties

We mention several important properties of isogenies.

1. Isogenies are uniquely determined by their kernel: Given an elliptic curve $$E$$ and a subgroup $$L$$, there is an unique elliptic curve $$E'$$ and an isogeny $$\alpha: E \rightarrow E'$$ such that the kernel of $$\alpha$$ is $$L$$.

2. [Sil09, Chapter III.6, Theorem 6.2] For every isogeny $$\alpha: E \rightarrow E'$$ of degree $$l$$, there exists an unique dual isogeny $$\hat{\alpha}: E' \rightarrow E$$ such that $$\alpha \hat{\alpha}= \hat{\alpha} \alpha=l$$

3. [Gha21, Proposition 2.2] (Decomposition of isogenies) Let $$\alpha: E \rightarrow E'$$ be a seperable isogeny. Then there exists an integer $$k$$ elliptic curves $$E=E_0, E_1,...,E_n=E'$$ and isogenies $$\beta_i: E_i \rightarrow E_{i+1}$$ of prime degree such that $$\alpha=\beta_{n-1} \beta_{n-2} ... \beta_{0} [k]$$

The edges of Supersingular isogeny graphs are determined by isogenies between the curves, we will talk about it later in the definition of the graph.