# How did Weil prove the Weil conjectures for curves?

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!

• Rationality and functional equation were proved by F. K. Schmidt before Weil. It only needs Riemann-Roch and you can find it in many books. I don't think Weil produced a new proof of these. The proof in your link is the classical proof rewritten in modern language. Jul 28, 2013 at 16:05
• To follow up on Felipe's comment, please look at Peter Roquette's paper at rzuser.uni-heidelberg.de/~ci3/manu.html#Class for a discussion of the work in the 1920s and 1930s on the zeta-function of function fields over finite fields. It includes a treatment of Schmidt's work. Aug 1, 2013 at 6:57

## 2 Answers

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.

• Stepanov "only" proves the RH part. OP was asking about rationality and functional equation. Jul 28, 2013 at 18:52
• @FelipeVoloch Fixed, I hope... Jul 28, 2013 at 19:05
• The proof in your link is the same proof as in the OP's link and is the same proof as in F.K. Schmidt, Analytische Zahlentheorie in Körpern der Charackteristic p, Math. Z. 33 (1931), 1–32. Jul 28, 2013 at 19:51
• Excepting the proofs that apply to higher dimensions (Dwork, Grothendieck,...) I only know one proof of rationality and functional equation. That one. Break the Dirichlet series of the zeta function in two sums according to degree $\le 2g -2$ or not, sum the geometric series for the infinite sum (this already gives rationality) and use Riemann-Roch (or Serre duality, whatever you want to call it) on the finite sum to relate a divisor $D$ with $K-D$ ($K$ a canonical divisor) and deduce the functional equation. You can talk about divisors, closed points, ideals, prime ideals, valuations, etc... Jul 28, 2013 at 20:06
• It's all the same. Jul 28, 2013 at 20:06

This syllabus proves the Weil conjectures for curves with only basic notions of algebraic geometry. Schemes aren't necessary.