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I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this:

Let $\mathbb{k}$ be an algebraically closed field and $X$ be a smooth projective curve over $\mathbb{k}$. Let $\omega$ be a rational differential form on $X$. Then,

$\displaystyle\sum_{P \in X}^{} Res_P(\omega) = 0$

Here $P$ denotes the points on the curve $X$ and $Res_P(\omega)$ denotes the residues of $\omega$ at points $P$.

I know that this can easily be derived with the help of Green's/Stoke's theorems in case where $\mathbb{k} = \mathbb{C}$ , as $X$ will then have the natural structure of a Riemann surface and things like integrals will start making sense. But, I am unable to prove it for an arbitrary algebraically closed field $\mathbb{k}$ other than $\mathbb{C}$.

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You're right that this is not obvious. It is a major theorem. The two main approaches are by Serre: Groupes algebriques et corps de classes chapter 2, or Tate: Residues of differentials on curves (can be found on Numdam). –  Matt Oct 10 '12 at 16:44
    
@Matt I guess that's what I am looking for; Thanks. –  Ashu Pachauri Oct 11 '12 at 8:13

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