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The problem of calculating Hasse--Weil zeta-function for a given curve $C/\mathbb{F}_p$ over a finite field is far from being easy, especially for large genus (as discussed by Wouter Castryck at https://homepages.warwick.ac.uk/~maseap/arith/notes/pointcountingwarwick.pdf).

  1. I'm interested in examples of explicit calculations of Hasse--Weil zeta-functions for curves. Calculations made with the use of heavy machinery, such as $l$-adic cohomology, and calculations made with a more elementary number theory are welcomed.

  2. There is a connection between cocycles classes in the first cohomology group $H^1(C)$ and zeroes of Hasse--Weil zeta. Is there any sources where this connection is illustrated with explicit examples?

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For #1: Chapter 4 of Moreno's "Algebraic Curves over Function Fields" have a number of good explicit examples via exponential sums + geometry, including Artin-Schreier coverings. There are also good end-of-chapter exercises.

I'm not quite sure what you meant by #2: The reciprocal roots of the (numerator) of the zeta function of a curve are the eigenvalues of Frobenius action on the first etale cohomology of the curve --- is that what you have in mind?

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  • $\begingroup$ @W Sao. Thanks for the reference for #1! It was useful. What concerns #2, I had a feeling after reading some article quite long ago that there exists more or less canonical map taking each zeta's zero to some particular basis element of $H^1$. I wasn't interested in it then, so hadn't saved the reference, and searching doesn't help now. Probably, that was a wrong feeling. I think, I shall ask more precise question, or edit this one, later. $\endgroup$ – Yauhen Radyna Feb 7 '15 at 19:09
  • $\begingroup$ This should be Moreno's "Algebraic Curves over Finite Fields". $\endgroup$ – Watson Jun 4 '18 at 14:59

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