For Variety $V$, function $V\rightarrow \overline{K}$

Question

I’m reading Silverman’s Arithmetic of Elliptic Curves, and they make the comment “Since an element $f∈\overline{K}[V]$ is well-defined up to a polynomial vanishing on $V$, it induces a well-defines function $f:V\rightarrow \overline{K},$ where $V$ denotes an affine algebraic variety. My question:

Let’s start off by using more pedantic notation. Let $\overline{f}\in\overline{K}$, and let $\overline{F}$ denote this induced map $V\rightarrow \overline{K}$. How is $\overline{F}$ defined?

Thank you much.

Answer 1

Do you understand how the ring $\Bbb{C}[x,y]/(y^2-x^3-1)$ is constructed ? Its elements are functions from the elliptic curve $E=\{ (a,b)\in \Bbb{C}^2,b^2-a^3-1=0\}$ to $\Bbb{C}$ in the natural way $$(\sum_{i,j} c_{i,j}x^i y^j )(a,b) = \sum_{i,j} c_{i,j}a^i b^j$$ The main concept to investigate is that we can evaluate any polynomial of $\Bbb{C}[x,y]$ on $E$, then the ideal $(y^2-x^3-1)$ is exactly the subset of those vanishing on $E$.