I understand that Weil proved the Weil conjectures for curves. I have seen his proof of the third and trickiest part, the "Riemann Hypothesis for curves," but I am curious about how he showed rationality and the functional equation. These are relatively elementary in modern scheme-theoretic language, which was unavailable to Weil – see Sam Raskin Weil conjectures for curves. In particular, I am not sure how to cast the proof at this link into the classical language of varieties – even the definition of the zeta function given there, as a product over the closed points of $X/\mathbb{F}_q$ seems hard to translate. (I know that you could just define it by the exponential generating functional, but then what kind of product formula could you prove?)
In summary, I would like to see an outline/sketch of a classical approach to the first two parts of the Weil conjectures for curves, especially Weil's own proof!
Best Answer
Can't speak for Weil, but a very nice writeup of the more elementary Stepanov approach to Weil's theorem was done by Ariel Gabizon (together with Avi Widgerson and Zeev Dvir, I think), to be found here.
Edit, to meet Felipe's objection
Rationality/functional equation is proved here.