Weierstrass form for toric elliptic curves

[vbraun]

This ticket implements the Weierstrass form (of the Jacobian) of anticanonical hypersurfaces in toric surfaces defined by reflexive polygons:

  sage: from sage.schemes.toric.weierstrass import WeierstrassForm
  sage: R.<x,y> = QQ[]
  sage: cubic = x^3 + y^3 + 1
  sage: WeierstrassForm(cubic)  # cubic in P^2
  (0, -27/4)
  sage: WeierstrassForm(x^4 + y^2 + 1)  # in P^2[112]
  (-4, 0)
  sage: WeierstrassForm(x^2*y^2 + x^2 + y^2 + 1)   # in P^1xP^1
  (-16/3, 128/27)

Apply

• trac_13084_ppl_lattice_polygon.patch​
• trac_13084_toric_weierstrass.patch​
• trac_13084_weierstrass_cleanup-fc.patch​
• trac_13084_ppl_cleanup-fc.patch​
• trac_13084_misc.patch​

[nbruin]

Excellent work! In all these cases, the genus 1 curve is a cover of its jacobian in a relatively canonical way. With your approach, is it reasonably possible to compute these covering maps? Doing so gives a kind of "certificate". One starting point would be

T.A Fisher, The invariants of a genus one curve, Proc. Lond. Math. Soc., 97(3):753--782, 2008.

also on ​arXiv.

[vbraun]

The reference looks interesting, I'll have a look!

Though I think there is already too much code in this ticket, any further features should go to a followup.

[vbraun]

I've split off the classical invariant theory stuff into a different module (#13451), this is much cleaner now. Still needs review ;-)

[chapoton]

apply only trac_13084_toric_weierstrass.patch

[vbraun]

rediffed for sage-5.8.beta0

[vbraun]

This is mostly geometry of polygons with a little bit of invariant theory thrown in - Dima, do you feel comfortable reviewing the ticket?

[dimpase]

Replying to vbraun:

This is mostly geometry of polygons with a little bit of invariant theory thrown in - Dima, do you feel comfortable reviewing the ticket?

This is surely a fun ticket!

But I am not familiar with the number theory part of it, and lately falling behind with everything, including reviewing my son's nappy's :–) Besides, it ought to be reviewed by number theorists.

Let me in turn invite you to consider taking part in this: ​http://web.spms.ntu.edu.sg/~dima/IMS2013/

[vbraun]

For the record, there isn't really any number theory in here. Maybe Marco is interested in reviewing this? ;-)

[chapoton]

See the bot report:

• coverage is not 100%

• needs to use the new doctest continuation ...:

[vbraun]

Updated patch

[vbraun]

Updated patch

[vbraun]

Fixed

[vbraun]

Frédéric, are you going to review this ticket?

[chapoton]

I have added a "cosmetic cleanup" patch for the file weierstrass.py. In particular, it removes many unused import statements.

[vbraun]

Thanks, looks good to me.

[chapoton]

Another cosmetic-cleanup patch. Similar work on import statements and pep8

[vbraun]

Looks good to me.

I take it you forgot to set the ticket to positive review after posting patches for all of the new code?

[chapoton]

well, I am not able to seriously review the math..

[dimpase]

Attempting to look at the maths side. It's confusing that you start off talking about a toric elliptic curve, but then speak about "the elliptic curve". What is going on there?

further nitpicks:

There are 16 reflexive polygons in 2-d.

Could you add a reference and change this to

There are 16 reflexive polygons in the plane.

And add an exact reference to the next statement.

Each defines a toric fano  variety, which (in 2-d) has a unique crepant resolution to a smooth 
 toric surface.

Which should also be changed to

Each of them defines a toric Fano  variety...

Then,

It turns out that the anticanonical hypersurface equation...

Reference?

[vbraun]

Replying to dimpase:

And add an exact reference to the next statement.

Each defines a toric fano  variety, which (in 2-d) has a unique crepant resolution to a smooth 
 toric surface.

Thats just the fact that a fan in the plane has a unique refinement into smooth cones. CLS just say that "its easy to see" (p. 500)

[vbraun]

Done

[dimpase]

Replying to vbraun:

Done

capitalize fano. It's a ​name, after all!

[dimpase]

hope you will update the relevant patch to change fano to Fano in docstrings...

[vbraun]

Updated patch

[vbraun]

Thanks, I fixed the "Fano".

Speaking of, its been grinding my gears the last couple of days that hilbert_series is lower case in Sage ;-)