QuadFormWithIsom: further cleanups + preparation for incoming new features

[StevellM]

Here are some further minor changes for the package:

• Clean-up of the docstrings and of the html documentation (I was not happy with how it looked);

  • Fix the different bugs in display, the typos and other mistakes
  • Add more internal references to different part of the html documentation
  • Add more descriptions/comments for the internal functions

• More improvements following the previous patch:

  • Disable more unnecessary checks
  • Avoid the creation of some lists in largely used functions when it can be replace by "smarter math"
  • Avoid to use cartesian_product_iterator for products of two lists, and put back inplace = true for products of multiple lists (we might save more time and memory now)
  • Rename some variables which should not have the same type in a same code

• Few more functionalities for the soon to be added complement on equivariant primitive embeddings:

  • Recognition of primary, elementary and unimodular lattices with isometry
  • Computation of orthogonal submodules (whenever it makes sense)
  • Computation of the representation of a group of isometries for a lattice on the discriminant group of that lattice
  • image_centralizer_in_Oq extended to the general case using hermitian Miranda-Morrison and gluing of stabilizers along equivariant primitive extensions
  • Make the function primitive_extensions part of the available functions and not only a background routine (to be safe I have added a sort of deprecations)

[codecov]

Codecov Report

Merging #2847 (593d3bc) into master (aa8e64f) will decrease coverage by 0.06%.
Report is 26 commits behind head on master.
The diff coverage is 97.92%.

@@            Coverage Diff             @@
##           master    #2847      +/-   ##
==========================================
- Coverage   80.65%   80.60%   -0.06%     
==========================================
  Files         456      456              
  Lines       64729    65411     +682     
==========================================
+ Hits        52209    52723     +514     
- Misses      12520    12688     +168     

Files	Coverage Δ
experimental/QuadFormAndIsom/src/enumeration.jl	96.02% <100.00%> (+<0.01%)	⬆️
...mental/QuadFormAndIsom/src/spaces_with_isometry.jl	98.36% <ø> (ø)
experimental/QuadFormAndIsom/src/types.jl	100.00% <ø> (ø)
experimental/QuadFormAndIsom/test/runtests.jl	97.29% <ø> (ø)
...ntal/QuadFormAndIsom/src/lattices_with_isometry.jl	98.94% <98.23%> (+2.34%)	⬆️
experimental/QuadFormAndIsom/src/embeddings.jl	98.04% <97.53%> (+0.16%)	⬆️

... and 56 files with indirect coverage changes

[fingolfin]

Looks good to me. Perhaps @simonbrandhorst should also approve and/or merge it?

[simonbrandhorst]

I am not sure this statement is true. Can you give a reference?

[StevellM]

I might be wrong but I guess this follows from Nikulin. If you have fixed isometry class of subgroups $H_M$ for $D_M$ and $H_N$ for $D_N$ which are anti-isometric, then the isometry class of the discriminant form for the overlattice is independent on the choice of the representatives in the isometry classes of $H_M$ and $H_N$. Or am I using a wrong shortcut ? Otherwise I could just revise the documentation to avoid this statement.

[simonbrandhorst]

As discussed, this part of the code needs to be changed to be correct.

[StevellM]

There is more than what was originally planned: we remarked that the code for primitive_embeddings was not totally right for some cases that were not reached. It has been modified accordingly.

[fingolfin]

What's the status here? Is this ready now, or is more work needed?

[StevellM]

For me it is good to go, maybe @simonbrandhorst would like to check whether my fixes are correct.