## 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 $$ = C_O,  = id, [m] = [m-1] +  (m \gt 1), [m] = -[-m] (m \lt 0)$$.

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 $$1$$.

Ben Lynn blynn@cs.stanford.edu 💡