Let $C$ be the algebraic curve defined by the modular polynomial $\phi_N$ of order $N>1$ over the rational numbers, i.e. \begin{equation}C:=\text{specm}(\mathbb{Q}[X,Y]/\phi_N(X,Y)). \end{equation} The singularities of this curve can be removed and we obtain a nonsingular curve $C^{sn}$, then, we can embed $C^{sn}$ into a complete non-singular curve $\overline{C}$.
In Milne's notes "Modular Functions and Modular Forms" it is written:
The coordinate functions $x$ and $y$ are rational functions on $\overline{C}$, they generate the field of rational functions on $\overline{C}$, and they satisfy the relation $\phi_N(x,y)=0$.
I assume, by coordinate functions he means the functions $f(X),g(X)\in \mathbb{Q}[X]$ such that $\phi_N(f(x),g(x))=0$ for all $x\in Q$. However, I don't understand why the field of rational functions on $\overline{C}$ is generated by these functions. Could someone explain this to me?
Thank you very much in advance!
The coodinate functions $x$ and $y$ are the image of $X$ and $Y$ in the quotient of $\mathbb{Q}[X,Y]$. The map $\xi: \mathbb{Q}[X] \rightarrow Q[X,Y]/(\phi_N(X,Y))$ obtained from the composition $\mathbb{Q}[X] \hookrightarrow \mathbb{Q}[X,Y] \twoheadrightarrow Q[X,Y]/(\phi_n(X,Y))$ will correspond to the map $x: C \rightarrow \mathrm{Spec} \mathbb{Q}[X]=\mathbb{A}^1_\mathbb{Q}$, which is a regular function on the non-singular locus $C^{sn}$ of the completed curve $\overline{C}$, in other words a rational function.
That $x$ and $y$ satisfy the relation $\phi_n(x,y)$ is just by definition.
The injection $\mathbb{Q}[X] \hookrightarrow \mathbb{Q}[X,Y]$ corresponds to the projection $\mathbb{A}^2_\mathbb{Q} \rightarrow \mathbb{A}^1_\mathbb{Q}$ onto the x coordinate, assuming we identify the points of $\mathbb{A}_\mathbb{Q}^2$ with pairs $(x,y)$. If $C$ is a curve living in $\mathbb{A}^2_\mathbb{Q}$, the map above is just its restriction to $C$, so it's the map giving the $x$ coordinate of the points on $C$.