Field of Rational Functions

Let $E(K)$ be an elliptic curve with equation $f(X, Y) = 0$ [the following is true for any affine curve].

Take a polynomial $g(X, Y)$, and consider its behaviour on the points of $E(K)$ only, ignoring its behaviour on all other values of $X$ and $Y$.

Then, for example, if $g = f$, from our point of view, $g$ is the same as the zero function because $g(P)$ for any point $P$ on the curve is zero. In fact, it can be shown (using Hilbert’s Nullstellensatz) that a polynomial $g$ is the zero function on $E$ if and only if it is a multiple of $f$.

This leads us to define the ring of regular functions of $E$ to be

\[K[E] = K[X,Y]/\langle f \rangle \]

Its field of fractions $K(E)$ is called the field of rational functions of $E$.

If we write $E$ in Weierstrass form, then we can always replace $Y^2$ with smaller powers of $Y$ meaning that a regular function can always be written in the form $v(X) + Y w(X)$.

Fact: Every nonconstant regular function has at least two finite zeroes.