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?


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?

| cite | improve this answer | |
  • $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.