# Explicit examples of Hasse--Weil zeta-function calculations for curves

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?

• @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. – Yauhen Radyna Feb 7 '15 at 19:09