Add style guide / reference for # optional - sage.... doctest tags, extend sage -t and sage -fixdoctests for modularization tasks

[mkoeppe]

📚 Description

• Fixing the handling of file-level # optional tags.
• Some files were not being doctested; fixing the recently introduced errors in doctests.
• Implementing block-scoped tags (originally done in PR Implement persistent optional tag  #35779, merged here)
• Expanding the documentation on the # optional / # needs tags used for modularization purposes, with examples.
• Adding features sage.modular, sage.numerical.mip, sage.rings.complex_double, sage.sat
• The tools sage -t and sage --fixdoctests receive some new options for modularization tasks (see added documentation):

  $ make pypi-wheels
  $ make SAGE_CHECK=yes sagemath-modules
  $ ./sage --fixdoctests --distribution sagemath-modules \
      src/sage/combinat/root_system/root_lattice_realization_algebras.py
  

  (example uses Modularization of sagelib: Distributions sagemath-{brial,combinat,eclib,flint,gap,giac,glpk,graphs,groups,homfly,lcalc,libbraiding,libecm,linbox,modules,mpmath,ntl,pari,plot,polyhedra,schemes,singular,standard-no-symbolics,symbolics} #35095)
• Suppressing # optional/# needs of present features in the help system
  .

Motivated by the sage-devel thread https://groups.google.com/g/sage-devel/c/utA0N1it0Eo (2023-06)

• Resolves Doctest # optional / # long / # needs annotation maintenance tool #35790
• Resolves Codeblock-scoped # optional annotations for doctests (modularization) #35750
• Part of Meta-ticket: Modularize sagelib into separate distributions (pip-installable packages) sagemath-... for Sage 10.x #29705

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

• Depends on sage.features: Declare features as "standard" explicitly #35820 (merged here)
• Vote at https://groups.google.com/g/sage-devel/c/8KZNRBs6F6U (result: https://groups.google.com/g/sage-devel/c/MtS2u3VbJEo)

[mkoeppe]

(the doctest failure in ell_rational_field is unrelated)

[mkoeppe]

Thank you!

[vbraun]

Doctests fail because now the previously-skipped broken files are picked up

sage -t --long --random-seed=286735480429121101562228604801325644303 sage/schemes/elliptic_curves/heegner.py  # 4 doctests failed
sage -t --long --random-seed=286735480429121101562228604801325644303 sage/schemes/elliptic_curves/hom_frobenius.py  # 2 doctests failed

[kwankyu]

From sage-release

That's already fixed in #35749, which is why that PR is marked critical.

We didn't fix the failures. Just that @mkoeppe mentioned that they are not related with this PR. According to @vbraun, indeed they are.

[kwankyu]

They all look easy to fix. All the doctests rely on arbitrary choice between $P$ and $-P$ which are points of an elliptic curve whose $x$-coordinates are the same. I am 100% sure about the doctests in age/schemes/elliptic_curves/hom_frobenius.py and 80% sure about the doctests in sage/schemes/elliptic_curves/heegner.py. Needs a look from an expert @yyyyx4.

[kwankyu]

We may fix them in the style

--- a/src/sage/schemes/elliptic_curves/heegner.py
+++ b/src/sage/schemes/elliptic_curves/heegner.py
@@ -47,8 +47,8 @@ We find some Mordell-Weil generators in the rank 1 case using Heegner points::
     sage: E = EllipticCurve('43a'); P = E.heegner_point(-7)
     sage: P.x_poly_exact()
     x
-    sage: P.point_exact()
-    (0 : 0 : 1)
+    sage: p = P.point_exact(); p == E(0,0,1) or -p == E(0,0,1)
+    True
 
     sage: E = EllipticCurve('997a')
     sage: E.rank()

[kwankyu]

Please merge #35966 here for test.

[github-actions]

Documentation preview for this PR (built with commit 44549f6; changes) is ready! 🎉