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20264 Hasse-Weil Zeta function of a cyclic cover of P1 over finite fields. edgarcosta "We add a new method to compute the zeta function of a cyclic cover of {{{P^1}}}, this is the result of a forthcoming paper generalizing the work of Kedlaya, Harvey, Minzlaff and Gonçalves.
In particular, we add two classes for cyclic covers, one over a generic ring and a specialized one over finite fields.
This requires wrapping David Harvey's code for computing products of matrices already in Sage but not accessible to Sage, see #25366
Here is a quick example:
{{{
sage: p = 4999;
sage: x = PolynomialRing(GF(p),""x"").gen();
sage: C = CyclicCover(5, x^5 + 1)
sage: C
Cyclic Cover of P^1 over Finite Field of size 4999 defined by y^5 = x^5 + 1
sage: C.frobenius_polynomial()
x^12 + 29994*x^10 + 374850015*x^8 + 2498500299980*x^6 + 9367502249700015*x^4 + 18731257498500149994*x^2 + 15606259372500374970001
sage: C.genus()
6
}}}" enhancement closed major sage-9.0 algebraic geometry fixed days71, sd87, days88 alexjbest Vishal Arul, Edgar Costa, Richard Magner, Nicholas Triantafillou Frédéric Chapoton N/A 612fca550e9bef381f5e8b447f71bbe5175a345f 612fca550e9bef381f5e8b447f71bbe5175a345f #25366