Elliptic Curves

A rational map of an elliptic curve $E(K)$ to itself is an element of the elliptic curve $E(K(E))$. (Recall $K(E)$ denotes the function field of $E$.)

Let us examine what this means. Suppose $(g_x,g_y)\in E(K(E))$, so that $g_x(X,Y),g_y(X,Y)$ are rational functions that happen to satisfy the equation for $E$. Then if $P=(x,y)$ is a point on the elliptic curve $E(K)$, we see that $(g_x(x,y),g_y(x,y))$ is also a point on an elliptic curve $E(K)$. So a rational map describes how to map points of $E(K)$ to other points of $E(K)$.

Fact: A rational map is either surjective or constant

A rational map is called an endomorphism or an isogeny if it maps the point $O$ to itself. The set of all endomorphism is denoted $End(E)$.

It is not hard to check that $End(E)$ is a ring (where addition is point addition, and multiplication is composition).

This is the pair of rational functions $X$ and $Y$.

It is also the multiplicative identity of $End(E)$.

Let $P$ be a point $(x,y)$. Then the constant map that takes every point to $P$ is given by

(both rational functions are constants). The map $C_O$ is only constant isogeny, and it is the additive identity of $End(E)$.

Let $P$ be a point $(x,y)$. Then the translation map that adds $P$ to every point is

Here the addition refers to point addition done over $E(K(E))$. The map can be explicitly given as follows. Set $\lambda$ to the rational function $(Y-y)/(X-x)$. Let

and

Then ${\tau }_P=(g_x,g_y)$.

The multiplication-by-$m$ map $[m]$ takes a point $P$ to $\mathrm{mP}$, and hence it can be defined as follows $[0]=C_O,[1]=\mathrm{id},[m]=[m-1]+[1](m>1),[m]=-[-m](m<0)$.

Explicit descriptions of the rational functions involved are usually given in terms of division polynomials (TODO).

Suppose we are working with a curve $E$ over ${𝔽}_q$. The Frobenius map is given by

It is easily verified that this is indeed an isogeny.

Certain curves have useful maps often called twists. In the following section $p$ denotes a prime.