For an algebraic curve in complex projective space, it's not too hard for me to develop a non-singular model as a Riemann surface and then use analytic techniques to show that an algebraic (read meromorphic) function is completely characterized by its divisor (up to a multiplicitive constant) or the principal parts of its series expansions at its poles (up to an additive constant).
I'm pretty sure that this is also true if we drop the assumption that we're working over the complex field. All we need is projective space for these two theorems to be true.
Can somebody point me to a reference where this is shown?
Hartshorne Corollary 6.10. A principle divisor on a complete nonsingular curve has degree zero.
This corollary almost does the trick for divisors. Just need to show that the constant functions are the only functions with no poles or zeros.
Again, this isn't hard (for me) analytically. I'm trying to figure how to do it algebraically, over fields that might not be algebraically closed or have characteristic zero.
The key observation is that the only regular functions on a projective variety are the constants. This is Theorem I.3.4(a) of Hartshorne, but Hartshorne's entire book is based on an algebraically closed base field. Looking at the proof, I think we can drop algebraic closure, at least for this theorem.
Given two rational functions with the same principal parts, we can subtract them from each other. The principal parts cancel, so we're left with a regular function, which must be constant. Thus, principal parts determine a rational function up to an additive constant.
Given two rational functions with identical divisors, we can divide one by the other and the resulting function's divisor will be zero, since $(f/g) = (f) - (g)$ (Hartshorne p. 131). Again, this is a regular function which must be constant, so divisors determine a rational function up to a multiplicative constant.
Hartshorne's treatment of divisors is based on the assumption that we're working on a "noetherian integral separated scheme which is regular in codimension one". Integral implies irreducible. For a curve, "regular in codimension one" means non-singular. Without this condition, we can't even cleanly define the divisor of a function, since a function could have a pole on one component of a singularity and a zero on another.
So, the theorems are true on non-singular irreducible projective curves, with no restriction on the characteristic.