## 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.

Ben Lynn blynn@cs.stanford.edu 💡