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

[git]

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

​dd82e8f

Merge branch 'public/zeta-no-wrap' of git://trac.sagemath.org/sage into HEAD

[git]

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

​b5d81b9

fix deprecation, more idiomatic

[git]

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

​f342349	Merge branch 'develop' of git://github.com/sagemath/sage into HEAD
​b707b2d	Merge branch 'develop' of git://github.com/sagemath/sage into HEAD
​4a10de1	fixing bugs, adding tests, and some whitespace
​e215187	Merge branch 'public/zeta-no-wrap' of git://trac.sagemath.org/sage into t/20264/zeta

[edgarcosta]

The bugs mentioned before have been fixed

[chapoton]

• Use "Return" and not "Returns" in the first line of documentations ; same for "Computes", etc

• Do not end the lines by \ when using ....:, they are not needed

• the \ is also not needed to wrap lines inside the text of an Error, just write something like

raise BeautifulError("all the carrots "
                     "belong to the rabbits")

• Use https://www.gnu.org/licenses/ in the headers (with https instead of http)

• Capital F to Frobenius in the documentation

• Is the change to src/sage/schemes/hyperelliptic_curves/hypellfrob.pyx needed ?

• The file src/sage/schemes/cyclic_covers/__init__.py should rather be empty

• src/sage/schemes/cyclic_covers/charpoly_frobenius.py:8: 'sage.matrix.constructor.matrix' imported but unused

• When testing that something mod 2 is not zero, you can write if k % 2:

ok, enough for the moment

[chapoton]

and many more tests should be tagged as #long

the longest one takes too long and should remain # not tested, very long

[alexjbest]

Replying to chapoton:

• Use "Return" and not "Returns" in the first line of documentations ; same for "Computes", etc

• Do not end the lines by \ when using ....:, they are not needed

• the \ is also not needed to wrap lines inside the text of an Error, just write something like

raise BeautifulError("all the carrots "
                     "belong to the rabbits")

• Use https://www.gnu.org/licenses/ in the headers (with https instead of http)

• Capital F to Frobenius in the documentation

• Is the change to src/sage/schemes/hyperelliptic_curves/hypellfrob.pyx needed ?

• The file src/sage/schemes/cyclic_covers/__init__.py should rather be empty

• src/sage/schemes/cyclic_covers/charpoly_frobenius.py:8: 'sage.matrix.constructor.matrix' imported but unused

• When testing that something mod 2 is not zero, you can write if k % 2:

ok, enough for the moment

Most of these are fine and I am happy to do, but I wonder if you know an automatic tool to fix these things, if not perhaps we should make one, it would save time fixing annoying single character changes. As for k %2 I don't know why I would do that, to save keystrokes? It makes it less readable IMO and doesn't seem to have benefit.

As for the change to src/sage/schemes/hyperelliptic_curves/hypellfrob.pyx I'm not sure how that appeared there, maybe a merge issue, if they don't break anything I'll remove them I suppose.

[git]

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

​6e76b12

whitespace

[git]

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

​6818510

anglais

[git]

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

​6f438b9	white space
​c541299	white space

[chapoton]

Humm. not sure what you are doing, But I did not suggest anything about white space.

And you can write the Errors on two lines (not 4), exactly as I suggested before.

[chapoton]

Usually, one does not put spaces around powers, so x**n is fine.

If you want to know how you are supposed to write code exactly, you should use pycodestyle to check for pep8 correctness. This is not strongly required.

[git]

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

​d773a8c

fix long time markers

[git]

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

​cf148c3

chapotonify

[alexjbest]

I think we got all your suggestions now, thanks!

I'm still not sure which tests are too long now, sage -t already includes warn long and shows nothing on my laptop, is there a good way to determine this?

[chapoton]

Not so amused by your commit titles, sorry to say. Your code has to meet Sage's standards, that's just the way it goes. I am really trying to help.

More fixes:

• Typo "instegral"

• Is there some reference article, either as preprint or published ? If yes, should be added.

• One spurious empty line here

  +        r"""
  +
  +        Return if this curve is singular or not.
  

• INPUT: and OUTPUT: must be followed by empty lines

• Typo in the sentence Return the polynomial of defining the cyclic cover. : remove "of"

• Remain some traces of copy-paste:

  Return this HyperellipticCurve over a new base ring R.
  

and

String representation of hyperelliptic curves.

• you should probably use the existing "is_squarefree" method of polynomials and not re-invent it in "check_squarefree":

  sage: g=(x-1)^2
  sage: g.is_squarefree()
  False
  sage: x = polygen(QQ)
  sage: g=(x-1)^2
  sage: g.is_squarefree()
  False
  sage: x = polygen(Qp(5))
  sage: g = x^2 + 5
  sage: g.is_squarefree()
  True
  sage: x = polygen(GF(2))
  sage: g = x^2 + x + 1
  sage: g.is_squarefree()
  True
  

[git]

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

​a274638	' ** ' -> '**'
​546e7e6	__init__ is now empty
​3582a15	Frobenius and \
​b47dcec	minor editing
​aae5de6	Merge branch 'public/zeta-no-wrap' of git://trac.sagemath.org/sage into t/20264/zeta
​6c8bd5d	Merge branch 'public/zeta-no-wrap' of git://trac.sagemath.org/sage into t/20264/zeta

[git]

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

​3325fad	typo
​464a227	removing traces of copy and paste
​7a8e924	blank lines after IN/OUTPUT:

[alexjbest]

Not so amused by your commit titles, sorry to say. Your code has to meet Sage's standards, that's just the way it goes. I am really trying to help.

Sorry that's more me being lazy than making a point! I appreciate the comments.

---

[git]

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

​516d088	typo and bad copy and pasta
​20164be	check_squarefree

[edgarcosta]

Thanks for your comments. I think I have addressed most of your comments, with the exception:

• When testing that something mod 2 is not zero, you can write if k % 2:

which I disagree with

ok. That would be slightly faster.

• Is there some reference article, either as preprint or published ? If yes, should be added.

as the answer is yes, and I need to figure out how to add the reference properly.

The reference itself should be added in the master reference file:

src/doc/en/reference/references/index.rst

and called in your files somewhere using the usual syntax [Who2019]_

• long tests

How do we determine what is long?

using

./sage -bt --long --warn-long src/sage/schemes/cyclic_covers/

Well, your machine may be too quick. Mine is slow. I will launch this command and report.

I also have an honest question, do you know the reason for not using the 3rd person in the documentation, e.g., return instead of returns? I know it is pep8 stuff, but I am confused about how should I be reading docstrings for this to make sense grammatically, as I always read them as the function F does something...

This is not pep8 stuff, but sage's own decision. The first line of the documentation is supposed to be in imperative mode (like when giving an order) as explained here:

​http://doc.sagemath.org/html/en/developer/coding_basics.html#documentation-strings

[chapoton]

Here are some timings

 sage.schemes.cyclic_covers.cycliccover_finite_field.CyclicCover_finite_field.frobenius_polynomial
Warning, slow doctest:
    CyclicCover(11, PolynomialRing(GF(1129), 'x')([-1] + [0]*(5-1) + [1])).frobenius_polynomial() # long time
Test ran for 27.69 s
**********************************************************************
File "src/sage/schemes/cyclic_covers/cycliccover_finite_field.py", line 1122, in sage.schemes.cyclic_covers.cycliccover_finite_field.CyclicCover_finite_field.frobenius_polynomial
Warning, slow doctest:
    CyclicCover(3, PolynomialRing(GF(1009^2), 'x')([-1] + [0]*(5-1) + [1])).frobenius_polynomial() # long time
Test ran for 135.69 s
**********************************************************************
File "src/sage/schemes/cyclic_covers/cycliccover_finite_field.py", line 1131, in sage.schemes.cyclic_covers.cycliccover_finite_field.CyclicCover_finite_field.frobenius_polynomial
Warning, slow doctest:
    C.frobenius_polynomial()
Test ran for 26.42 s

I sugggest that at least the worst one become # not tested

[chapoton]

There is a line

....:)

that is missing a white space after :

[git]

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

​e0411bf	Merge remote-tracking branch 'origin/develop' into t/20264/zeta
​c871b10	remove empty line

[chapoton]

• The following

  +            min_val = min(
  +                [min([self._Qq(elt).valuation() for elt in row]) for row in F.rows()]
  +            )
  

could be done in one loop

min_val = min(self._Qq(elt).valuation() for row in F.rows() for elt in row)

• In this

  +        Compute p-adic Frobenius matrix to precision p^N. If N not
  +        supplied, a default value is selected, which is the minimum needed
  +        to recover the charpoly unambiguously.
  

the first sentence should be followed by an empty line.

• In all.py, do not use assert

• In this sentence

  +    - ``charpoly_prec`` -- a vector ai, such that, `frob_matrix.change_ring(ZZ).charpoly()[i]`
  +        will be correct mod `p^ai`, this can be easily deduced from the hodge numbers and
  +        knowing the q-adic precision of `frob_matrix`
  

Hodge take a capital H, and the final frob_matrix should be between double backticks

• You could replace

  import sage.schemes.curves.affine_curve as plane_curve
  

by

from sage.schemes.curves.affine_curve import AffinePlaneCurve

and then use that directly

• In src/sage/schemes/cyclic_covers/charpoly_frobenius.py, there are many divisions by 2. I would suggest to add from __future__ import division in order to make sure that the python3 behaviour will remain the same.

• this sentence does not make sense

  +    - ``f`` - univariate polynomial
  +      is not given, then it defaults to 0.
  

• do not use range(0, something) but range(something)

• still an empty line to be removed here:

  +    def _horizontal_matrix_reduction(self, s):
  +        r"""
  +
  

• this should not be indented

  +        OUTPUT:
  +
  +            A tuple of tuples ``( (D0, D1), (M0, M1) )``
  +            where MV_t  = M0 + t * M1 and DV_t = D0 + t * D1
  +
  

ok, enough for today

[git]

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

​e809210

fix doctest

[git]

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

​a5d84ac

more assorted fixes

[chapoton]

You should call the added reference somewhere using [ABCMT2019]_, say in charpoly_frobenius

[chapoton]

This is wrong

import sage.schemes.curves.affine_curve import AffinePlaneCurve

The first "import" should be "from" instead.

[git]

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

​75b3e46	typo import->from
​4030d27	move __future__ import
​b1e0f99	tests
​20b3d04	one more test and a try in charpoly
​7755bb0	better wording

[chapoton]

+REFERENCES::

should end with a single colon, and be followed by an empty line.

Once done, I will give a positive review, if you set to "needs_review". It is better to make further changes in later tickets.

[git]

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

​612fca5

formatting references

[edgarcosta]

Thank you!

[chapoton]

my pleasure