add missing EllipticCurveHom methods to Îlu isogenies

[yyyyx4]

The EllipticCurveHom_velusqrt class introduced in #34303 inherits from EllipticCurveHom (#32388, #32502), but it doesn't yet implement all of the required methods. In this patch, we add them.

Diff without the dependency: sagemath/sagetrac-mirror@7326051...8cb0f29

Depends on #34410

CC: @defeo @JohnCremona @tscrim @kwankyu

Component: elliptic curves

Author: Lorenz Panny

Branch/Commit: 8cb0f29

Reviewer: Travis Scrimshaw, Kwankyu Lee

Issue created by migration from https://trac.sagemath.org/ticket/34614

[sagetrac-git]

Changed commit from 0a38cc5 to 726f2db

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

7a64e8c	Merge branch 'public/change_default_composition_for_elliptic_curve_isogenies'
726f2db	add more EllipticCurveHom methods to EllipticCurveHom_velusqrt

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

7326051	Merge tag '9.8.beta1' into public/change_default_composition_for_elliptic_curve_isogenies
c6bbbc3	add more EllipticCurveHom methods to EllipticCurveHom_velusqrt

[sagetrac-git]

Changed commit from 726f2db to c6bbbc3

[tscrim]

comment:6

Code wise, basically looks good (although I would like the long doctest output lines broken up a little bit). Could someone check the math?

[tscrim]

Reviewer: Travis Scrimshaw

[sagetrac-git]

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

8e95731	Minor edits mostly on spaces
df1024d	Move as_morphism()

[sagetrac-git]

Changed commit from c6bbbc3 to df1024d

[kwankyu]

comment:8

I made minor edits.

I took this opportunity to move the method as_morphism() to the class EllipticCurveHom so that it works with √élu isogenies. The method may be useful to check elliptic curve morphisms, though it is reasonably fast only for small examples.

One remark: There are notes like

    def rational_maps(self):
        r"""
        Return the pair of explicit rational maps of this Îlu isogeny
        as fractions of bivariate polynomials in `x` and `y`.

        .. NOTE::

            The data returned by this method has size linear in the degree.

        EXAMPLES::  

The meaning of the note is obscure to me. How is the size of the data measured? If the data is a polynomial, then it may mean "the degree of the returned polynomial is linear in the degree of the isogeny". In this case, the data is two rational functions. You need to be explicit about what is the "size" of rational functions. I suggest that you rewrite the notes or remove them (as they are, the notes are not so useful).

[yyyyx4]

comment:9

Replying to Kwankyu Lee:

How is the size of the data measured?

It's literally the size of the object, as in "bytes of memory it takes to store".

The purpose of these remarks is to point out that merely writing down the result of calling these methods already requires time Ω(ℓ), which means the √élu speedup (complexity Õ(√ℓ)) cannot apply to them. Perhaps that point should be made clearer.

[kwankyu]

comment:10

Replying to Lorenz Panny:

Replying to Kwankyu Lee:

How is the size of the data measured?

It's literally the size of the object, as in "bytes of memory it takes to store".

Okay.

The purpose of these remarks is to point out that merely writing down the result of calling these methods already requires time Ω(ℓ), which means the √élu speedup (complexity Õ(√ℓ)) cannot apply to them. Perhaps that point should be made clearer.

I now see what you mean by the note. Then if you want to keep those notes, I suggest that you write this explanation (implication) for these notes, once and for all, at the head of the file hom_velusqrt.py. Non-experts need the explanation.

Otherwise I am positive to the branch.

[kwankyu]

Changed reviewer from Travis Scrimshaw to Travis Scrimshaw, Kwankyu Lee

[sagetrac-git]

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

8cb0f29

explain "output has linear size" remarks

[sagetrac-git]

Changed commit from df1024d to 8cb0f29

[yyyyx4]

comment:13

Thanks! I added a note about it as you suggested.

[kwankyu]

comment:14

Thank you!

[vbraun]

Changed branch from public/more_EllipticCurveHom_methods_for_EllipticCurveHom_velusqrt to 8cb0f29