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 💡