Generic zeta function method for schemes

[kedlaya]

Currently there is a generic method zeta_series for schemes over a finite field, but not zeta_function. It would be useful to have the latter so that zeta_series could invoke it when it is implemented (which hopefully will happen for hyperelliptic curves soon).

At the same time, also make zeta_series work over nonprime fields.

Component: algebraic geometry

Keywords: zeta function, scheme, finite field, sd87

Author: Ben Hutz

Branch/Commit: d47998c

Reviewer: David Ayotte

Issue created by migration from https://trac.sagemath.org/ticket/20308

[fchapoton]

comment:1

Here is a tentative

---

New commits:

d6e70db

using zeta_function to compute zeta_series if possible

[fchapoton]

Branch: public/20308

[fchapoton]

Commit: d6e70db

[roed314]

Changed keywords from zeta function, scheme, finite field to zeta function, scheme, finite field, sd87

[bhutz]

comment:3

It's not clear to me what the goal here is. The initial code seems to be using the zeta_function call to try and compute the zeta series as opposed to trying to compute zeta functions for generic schemes. So, are we improving zeta series or trying to make a generic .zeta_function()?? The later seems daunting...

I've improved the zeta_series code to allow for non-prime base fields.

[bhutz]

Changed branch from public/20308 to u/bhutz/20308

[bhutz]

New commits:

5e776a6

20308: improve zeta series for generic schemes

[bhutz]

Changed commit from d6e70db to 5e776a6

[kedlaya]

comment:5

Sorry, am on vacation this week and hence unable to participate in sd87, but I can still clarify what I had in mind here!

What I had in mind is that the generic .zeta_function() returns a NotImplementedError. Then the generic .zeta_series() method would check for the existence of a .zeta_function() method, and if present would call it and convert the rational function into a power series instead of falling back to manually counting points.

[kedlaya]

comment:6

Replying to @bhutz:

It's not clear to me what the goal here is. The initial code seems to be using the zeta_function call to try and compute the zeta series as opposed to trying to compute zeta functions for generic schemes. So, are we improving zeta series or trying to make a generic .zeta_function()?? The later seems daunting...

I've improved the zeta_series code to allow for non-prime base fields.

That is an important improvement, but it is actually separate from the issue I had in mind. Maybe we should create a new ticket and put this patch there instead?

[sagetrac-git]

Changed commit from 5e776a6 to 280e260

[sagetrac-git]

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

280e260

20308: add generic zeta_function()

[bhutz]

Author: Ben Hutz

[bhutz]

comment:8

ok, that is certainly reasonable to do. I've added the generic function and put my doc test is the right place.

My zeta_series improvement is such a minor change and related to the original functionality, I'm fine with leaving it as part of this ticket. Perhaps we should make a slight change to the ticket description to include improving zeta_series in the generic case?

[sagetrac-git]

Changed commit from 280e260 to 7f1bfec

[sagetrac-git]

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

7f1bfec

20308: fix typo

[kedlaya]

comment:10

Replying to @bhutz:

ok, that is certainly reasonable to do. I've added the generic function and put my doc test is the right place.

My zeta_series improvement is such a minor change and related to the original functionality, I'm fine with leaving it as part of this ticket. Perhaps we should make a slight change to the ticket description to include improving zeta_series in the generic case?

Changed the description accordingly. Unfortunately I won't be able to review this before the end of the week, but all the better if someone else can do it...

[DavidAyotte]

comment:11

Everything seems right. I ran the doctest and all test passed. For some reason I had difficulty generating the reference manual, but the problem doesn't seems to come from this ticket. I got this error :

Error building the documentation.
Traceback (most recent call last):
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/runpy.py", line 174, in _run_module_as_main
    "__main__", fname, loader, pkg_name)
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/runpy.py", line 72, in _run_code
    exec code in run_globals
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/site-packages/sage_setup/docbuild/__main__.py", line 2, in <module>
    main()
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/site-packages/sage_setup/docbuild/__init__.py", line 1642, in main
    builder()
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/site-packages/sage_setup/docbuild/__init__.py", line 720, in _wrapper
    getattr(DocBuilder, build_type)(self, *args, **kwds)
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/site-packages/sage_setup/docbuild/__init__.py", line 104, in f
    runsphinx()
  File "/projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/site-packages/sage_setup/docbuild/sphinxbuild.py", line 215, in runsphinx
    raise exception
OSError: [schemes  ] /projects/da1818ed-996d-4de6-acc6-361415b7725d/Src/sage-david.ayotte/SageMath/local/lib/python2.7/site-packages/sage/schemes/affine/affine_space.py:docstring
of sage.schemes.affine.affine_space.AffineSpace_field.points_of_bounded_height:4: WARNING: citation not found: Doyle-Krumm

[sagetrac-git]

Changed commit from 7f1bfec to d47998c

[sagetrac-git]

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

d47998c

20308: add OUTPUT to zeta_function

[bhutz]

comment:13

make doc-clean
make doc

is able to build all docs.

[vbraun]

comment:15

Reviewer name...

[bhutz]

Reviewer: David Ayotte

[vbraun]

Changed branch from u/bhutz/20308 to d47998c