What is a function on a Line or a Curve?

Question

I am reading up on Pairings using Elliptic curves & all the texts talk about functions on a Curve.

I am finding it difficult to even figure out what they mean by "function on a curve" or "function on a line"

The equation of a line or a curve itself is in the form of a function, but I am unable to figure what is "function on a curve" or "function on a line".

Some examples.

In Mathematical Cryptography by Silverman,

Theorem 5.36. Let E be an elliptic curve.
(a) Let f and f' be rational functions on E.

Another text says, it will first introduce function on a line before going to function on a curve

We first give a gentle introduction to the theory of divisors by looking at examples of functions on the line before considering elliptic curves.

What exactly is a function on a line or a curve? And how are there multiple functions on a line or a curve? All I understand is one function which is associated with the curve or the line (the curve or line equation). It's very difficult to understand what are these multiple functions on a line or a curve.

Answer 1

For the purposes of elliptic curves and pairings with affine coordinates, functions are rational functions (ratios of two polynomials) in the two variables $X$ and $Y$ with coefficients in compatible fields. Curves are the set of points where a particular function is zero. Lines are curves where the underlying function is a polynomial of total degree 1. A function on a curve (usually the curve is defined by another function) is the set of values that the function takes on points of the curve i.e. the value of the function at places where the other function is zero.

For example if we work over the rational numbers and consider the function $C(X,Y)=Y^2-X^3+X-1$. This defines the elliptic curve $E:C(X,Y)=0$ which we might write as $E:Y^2=X^3-X+1$. Consider also the function $L(X,Y)=2X-Y-1$, this defines the line $\ell:L(X,Y)=0$ which we might usually write $\ell:Y=2X-1$. Although the functions are defined for all rational values of $X$ and $Y$, we can specialise to values that lie on curves. The function $C$ on the curve $E$ is zero everywhere, but the function $L$ takes more interesting values. Consider $L$ evaluated at the point $(5,-11)$ which lies on $E$. This is 20. Likewise we can talk about the function $C$ on the "curve" $\ell$ e.g. if we take the point $(7,13)$ which lies on $\ell$ we see $C(7,13)=-168$.

Clearly we can talk about many different functions defined on $E$ and not just $L$.

There are interesting relationships between a function $f$ on a curve defined by a function $g$ and the function $g$ on a curve defined by the function $f$. These start with the observation that the zeroes are shared. In particular the zeroes of $L$ on $E$ are $(0,-1)$, $(1,1)$, and $(3,5)$ which are also the zeroes of $C$ on $\ell$.