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

[edgarcosta]

New commits:

​3e1041d	wrap ntl_mat_ZZ_p functionality
​96e0783	wrap harvey's intervalproducts
​d91d387	merge w/develop
​5706cca	initial commit for cyclic covers

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7b51922

more examples at the top

[vdelecroix]

update milestone 8.3 -> 8.4

[alexjbest]

Last 10 new commits:

​7c86227	began reviewing zeta functions of cyclic covers
​05199c0	removed many semicolons
​91ddb8e	small fix
​e91fb3c	rename folder
​ed9eca9	finish move
​cdca59d	delete unused function
​df1216f	more style fixes last bit of move
​c4e7788	move check squarefree
​c0df121	add to main reference
​30d61b5	more cleanup

[edgarcosta]

Last 10 new commits:

​9b89e3a	removed many semicolons
​2493a73	small fix
​5a8afa8	rename folder
​3fb8aa1	finish move
​e5fe996	delete unused function
​198943f	more style fixes last bit of move
​cd05163	move check squarefree
​b78a04d	add to main reference
​3883347	more cleanup
​d7a885c	fixed lots of docs with edgar

[alexjbest]

This is looking pretty good now to me, so I intend to mark this as positive review once #25366 is reviewed.