Rational Maps

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 $\mathrm{End}(E)$.

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

Identity Map

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

\[ id = (X,Y) \]

It is also the multiplicative identity of $\mathrm{End}(E)$.

Constant Map

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

\[ C_P = (x,y) \]

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

Translation Map

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

\[ \tau_P = id + C_P \]

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

\[ g_x(X, Y) = \lambda^2 + a_1 \lambda - a_2 - X - x \]

and

\[ g_y(X, Y) = -a_1 g_x(X, Y) -a_3 -\lambda x_3 + \lambda x_1 - y_1 \]

Then $\tau_P = (g_x, g_y)$.

Multiplication Map

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

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

Frobenius Map

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

\[ \phi = (X^q, Y^q) \]

It is easily verified that this is indeed an isogeny.

Twists

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

  • $E: y^2 = x^3 + x$, $p = 3 \mod 4$, Twist on $E(\mathbb{F}_{p^2})$: $(-X, iY)$, where $i$ is a square root of $-1$.

  • $E: y^2 = x^3 + 1$, $p = 2 \mod 3$, Twist on $E(\mathbb{F}_{p^2})$: $(\zeta X, Y)$, where $\zeta$ is a nontrivial cube root of $1$.