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.

[edgarcosta]

Vishal Arul spotted some bugs.

sage: %time CyclicCover(11, PolynomialRing(GF(1129), 'x')([-1] + [0]*(5-1) + [1])).frobenius_polynomial()
CPU times: user 16.1 s, sys: 124 ms, total: 16.3 s
Wall time: 16.5 s
x^40 + 7337188909826596*x^30 + 3818873571673594702967652840943135*x^20 + 24687045654725446027864774006541463602997309796*x^10 + 11320844849639649951608809973589776933203136765026963553258401

So the discrepancy is in the t^20 coefficient -- Minzlaff's code says it is

20187877911930897108199045855206 = 6 * 1129^10

while the current code says it is

3818873571673594702967652840943135 = 6 * 1129^10 + 1129^11.

the difference is exactly 1129^{11. So it does look like some kind of precision error...}

My bet is that error is either in working precision or the code to lift characteristic polynomial to ZZ

Another bug, that seems to be related with precision when using the slow method

sage: CyclicCover(3, PolynomialRing(GF(1009), 'x')([-1] + [0]*(5-1) + [1])).frobenius_polynomial()
x^8 + 266*x^6 - 1474149*x^4 + 270809546*x^2 + 1036488922561
sage: CyclicCover(3, PolynomialRing(GF(1009**2), 'x')([-1] + [0]*(5-1) + [1])).frobenius_polynomial()
x^8 + 1045817316825941*x^7 + 546866930086505850133464526551*x^6 + 190640968494636702312197540239024924597536618*x^5 + 49843906537040069873392691186784704861320520036908736885312*x^4 + 194087947845988228526704383964086734259184677590058*x^3 + 566821515149604163130057985174959967417111*x^2 + 1103577471286158480607964750564981*x + 1074309286591662654798721

It looks like the frobenius polynomial for GF(1009) is correct, but it is not correct for GF(1009**2). Comparing with Minzlaff's code, the right frobenius polynomial for GF(1009**2) should be

x^8 + 532*x^7 - 2877542*x^6 - 242628176*x^5 + 4390163797795*x^4 - 247015136050256*x^3 - 2982540407204025062*x^2 + 561382189105547134612*x + 1074309286591662654798721