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Let $C_1\subset \mathbb{P}^{N_1}$ and $C_2\subset \mathbb{P}^{N_2}$ be two curves. Then a map $\phi:C_1\to C_2$ can be defined as


where each $f_i$ is a homogeneous polynomial. Such a map induces a map $\phi^*:K(C_2)\to K(C_1)$ given by $g\mapsto g\circ \phi$. What confuses me here is that the rings $K(C_i)$ are defined by Silverman to be the rings $K(C_i\cap \mathbb{A}^{N_i})$, so e.g. $K(C_2)$ is of the form $K[X_1,\ldots,X_{N_2}]/I$ and I have trouble seeing what $g\circ \phi$ should be for some $g\in K[X_1,\ldots,X_{N_2}]/I$, since $\phi$ has sort of "one component too much".

Is the idea here that we should essentially write $\phi$ in the form

$$\phi = [f_0/f_i,\ldots,1,\ldots,f_{N_2}/f_i]$$

where $i$ corresponds to our chosen embedding $\mathbb{A}^{N_2}\hookrightarrow U_i=\{X_i\neq 0\}\subset \mathbb{P}^{N_2}$? Now given some $g(X_1,\ldots,X_{N_2})\in K(C_2)$, we get

$$\phi^*g=g\circ \phi =?$$

How is this map supposed to be concretely represented? I know Hartshorne uses the representation where $K(C_2)$ is just quotients of homogeneous polynomials of the same degree, so the composition is pretty obvious. I'm trying to understand how to work with Silverman's definition.

EDIT: I think the idea here is to think of $K(C_2)$ as the ring generated by $X_0/X_i,\ldots,X_{N_2}/X_i$ and $K(C_1)$ as the ring generated by $Y_0/Y_j,\ldots,Y_{N_1}/Y_j$, so the polynomial $g$ is actually $g(X_0/X_i,\ldots,X_{N_2}/X_i)$. Then the map would be given by

$$\phi^*g = g\left(\frac{f_0(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)}{f_i(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)},\ldots,\frac{f_{N_1}(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)}{f_i(Y_0/Y_j,\ldots,Y_{N_1}/Y_j)}\right).$$

Can anyone confirm this? This does seem to coincide with Hartshorne's definition, though checking it would be quite a mess...

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Although Silverman's book is a rich source of information on elliptic curves, his foundations on algebraic varieties in Chapter 1 are not optimal in my view.
He introduces the basic material of algebraic and arithmetic geometry in 16 pages, including affine and projective varieties over non-algebraically closed fields and Galois actions, in an old-fashioned language difficult to understand if you don't know the material beforehand.

To come back to your question, the main point to understand is that $K(C_i)$ is a field and does not correspond to genuine functions defined on $C_i$( which are all constant: $K[C_i]=K$) but to rational functions , which have poles on $C_i$.
So, no, $K(C_i)$ is not the ring generated by the $X_0/X_i,...$.
Silverman defines a morphism $C_1\to C_2$ by restricting rational maps $\mathbb P^n\to \mathbb P^n$ and demanding that these maps be defined at every point of $C_1$. [It is disconcerting that at the beginning of §3 he embeds both varieties in the same $\mathbb P^n$]
You must be very careful with that definition:
For example if $C_1$ is the conic defined by $xz=y^2$ in $\mathbb P^2$, the rational map $f:C_1\to \mathbb P^1: (x:y:z)\mapsto (x:y)=(y:z) \;$ is regular everywhere but it is impossible to write it $f(x,y,z)=(f_1(x,y,z):f_2(x,y,z))$ with $f_1,f_2$ homogeneous polynomials of the same degree vanishing any no point of $C_1$. You must use two descriptions, as I did: $(x:y)$ and $(y:z)$.
This says that the statement in the very first sentence of your question requires a careful interpretation if you want it to be true.

For foundational material on varieties, I recommend Fulton's Algebraic Curves, freely and legally downloadable from here.

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