Improve morphisms between Drinfeld modules

[xcaruso]

We implement several methods for arbitrary morphisms between Drinfeld modules.
Especially, we implement the computation of norms and characteristic polynomials.

@ymusleh @DavidAyotte @kryzar

📚 Description

Here is a list of the most important added methods:

• hom: fast interface for creating morphisms between Drinfeld modules
• _composition_/inverse: composition and inverse of morphisms between Drinfeld modules
• is_isomorphic: check whether two Drinfeld modules are isomorphic either over the base field or over an algebraic closure
• norm: computation of the norm of a morphism between Drinfeld modules
• dual_isogeny: computation of the dual isogeny
• characteristic_polynomial/charpoly: computation of the characteristic polynomial of an endomorphism between Drinfeld modules

We also implement addition of morphisms and the action of the function field $A$; this gives the structure of left $A$-module on the set of morphisms between two given Drinfeld modules.

📝 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.

[DavidAyotte]

7c8861c: This is indeed much more faster than using basic j-invariants.

[DavidAyotte]

One feature that could be also implemented is the conjugate:

sage: A = GF(5)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1, 1])
sage: phi.conjugate(u)  # u nonzero
# return the Drinfeld module T |--> u^(-1) \phi_T u

[xcaruso]

One feature that could be also implemented is the conjugate:

sage: A = GF(5)['T']
sage: K.<T> = Frac(A)
sage: phi = DrinfeldModule(A, [T, 1, 1])
sage: phi.conjugate(u)  # u nonzero
# return the Drinfeld module T |--> u^(-1) \phi_T u

Sure.
It's the same than phi.velu(u) but it's probably better to have both.

[xcaruso]

@kryzar @DavidAyotte @ymusleh
It's time, I think, to start reviewing this ticket. Would you be willing to do this?

[ymusleh]

Not sure if I can act as a formal reviewer since I'm not part of the Sage community and haven't had a PR merged, but I'll look over the code and provide feedback.

[kryzar]

@kryzar @DavidAyotte @ymusleh It's time, I think, to start reviewing this ticket. Would you be willing to do this?

Yep.

[ymusleh]

Just some minor comments, otherwise I think this is good for a positive review.

[DavidAyotte]

Thanks David for your comments. I addressed all of them. Going through the code, I also noticed that inversions of general isomorphisms (which are not automorphisms) were not implemented. So I fixed this.

Thanks for the update. For my part, I am happy with the current state of this PR.

[ymusleh]

I'm also happy with the state of this PR.

[kryzar]

Did my review (thank you for your patience).

[xcaruso]

@kryzar
Thanks for your review. I think I adressed all you comments.

[kryzar]

Thank you very much @xcaruso for writing this contribution and taking the tame to address my comments! I approve the changes.

[xcaruso]

Thanks for your reviews.
I give a positive review to this PR; please revert it if you disagree.

[github-actions]

Documentation preview for this PR (built with commit 5f76571; changes) is ready! 🎉