Lattes to curve function code commits for enhancement #38329

[Nathabolin]

• Fixing issue Finding an Elliptic curve from a Lattes map #38329
• Adding Function with documentation to be reviewed for the above ticket
• Lattes to curve function

📝 Checklist

• The title is concise and informative.
• 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 and checked the documentation preview.

⌛ Dependencies

[bhutz]

A number of simple things to fix here:

• 6843: Finds a short Weierstrass model associated to this Lattes map
• 6844 - extra white space
• 6845 self is assumed to be
• 6848, 6852, 6853 self
• 6866, 6882, 6895 white space between arguments
• 6875 extra blank line
• 6911, 6920, 6928, 6934, 6942 fix indent
• 6914,6915,6916 extra white space
• 6918 extra \
• 6912-25, 6929 spacing needed

Less simple: I think you should add at least one doc tests that checks the output is correct. Something like

E, m = F.Lattes_to_curve()
P.Lattes_map(E,2).conjugate(m) == F

Finally, you cannot in general assume that u=1. The problem you are running into in the more general case is that the solution is not defined over QQ, not that there isn't a solution.

[bhutz]

I did a full functionality test and found a couple things. There are a couple more minor coding things too.

• need to wrap the very long output lines in the doctests
• and the defining polynomial of any required extension is not actually part of the output. That is just part of the elliptic curve definition, so remove that phrase.
• move imports to top of file
• 6939-40 These lines are not used

P = ProjectiveSpace(self.base_ring(), 1, "x, y")
 x, y = P.gens()

• I would suggest a parameter that defaults to true check_Lattes=True. If that parameter is true, then call is_Lattes after the degree check (6920). Add a test against a simple map

P.<x,y>=ProjectiveSpace(QQ,1)
f=DynamicalSystem([x^4,y^4])
f.Lattes_to_curve()

• There are some issues over number fields. The following fails due to a naming conflict on the variables

K.<a>=QuadraticField(2)
P.<x,y>=ProjectiveSpace(K,1)
E=EllipticCurve([1,a])
f=P.Lattes_map(E,2)
print(f)
f.Lattes_to_curve()

I would suggest renaming the variables to something no-one would typically use such as

    a = pts[0]['avar']
    b = pts[0]['bvar']
    u = pts[0]['uvar']
    v = pts[0]['vvar']
    t = pts[0]['tvar']
    w = pts[0]['wvar']

Nothing else needs to change for this since you use

a, b, u, v, w, t = R.gens()

but you do need to fix another coercion problem from this

eq += [R(g.coefficient({x:i})) - R(g_sym.coefficient({z:i}))]

Add a number field example after you fix these.

• Similarly there are problems coercing to QQbar when the field doesn't have the coercion specified

R.<t>=QQ[]
K.<c>=NumberField(t^3-2)
P.<x,y>=ProjectiveSpace(K,1)
E=EllipticCurve([1,c])
f=P.Lattes_map(E,2)
f.Lattes_to_curve()

This can be fixed by changing the code to

This can be fixed by changing the code to do:
phi = R.base_ring().embeddings(QQbar)[0]
eq = [poly.change_ring(phi) for poly in eq]
I = R.ideal(eq)
pts = I.variety()

• Finally: This is returning an error, but it should work:

P.<x,y>=ProjectiveSpace(QQbar,1)
E=EllipticCurve([1,2])
f=P.Lattes_map(E,2)
f.Lattes_to_curve()

so you just need to allow QQbar through in your check against numberfields

[Nathabolin]

"move imports to top of file"
I'm going to do this of course but oddly enough
the file already does try to import them

lazy_import('sage.rings.qqbar', 'number_field_elements_from_algebraics')

is on line 118

[bhutz]

I having all the functionality working in this version. I still have a couple comments though.

• The example
  """
  sage: P.<x,y> = ProjectiveSpace(QQbar,1)
  sage: F = DynamicalSystem_projective([x^4,y^4])
  sage: F.Lattes_to_curve(check_lattes=True)
  Traceback (most recent call last):
  ...
  NotImplementedError: Base ring must be a number field
  """
  is failing for is_Lattes not for Lattes_to_curve. I also don't see a reason why it doesn't allow QQbar. Why don't you make the same check in that function too.

• Since the default is not to check if its Lattes, maybe add a check
  """
  if len(pts) == 0:
  raise ValueError("No Solutions found. Check if map is Lattes.")
  """

• There are some white spacing issues in the later examples.

[vbraun]

Merge conflict

[Nathabolin]

@vbraun Could you point me to where to see the merge conflicts?

[vbraun]

Looks like you fixed the conflict in the last merge commit