Integrating the function fields implementation and with the curves implementation

[Paddy Adams]

At the moment the implementation of algebraic curves in Sage doesn't seem to be connected at all to the implementation of Function Fields. Is this intentional? It would seem to be useful to be able to, for example, get the function field from a curve. Also surely the Riemann-Roch basis function on projective curves should return elements of a function field and not, as it does currently, the field of fractions of k[X,Y,Z].

Thanks to the current global meltdown I am stuck at home with time on my hands so happy to try and tie together these two areas if there isn't a specific reason not to.

 

[John Cremona]

Certainly this is not intentional. It just has not been done. Having this done well (and efficiently) would be a major step towards bringing Sage closer to Magma's capabilities in this area.

+1

[Kwankyu Lee]

On Saturday, March 28, 2020 at 6:16:25 PM UTC+9, John Cremona wrote:

Certainly this is not intentional. It just has not been done.  

Wrong. It was recently done for curves over finite fields in affine and projective spaces.

As we already have function fields over number fields, it would be a rather trivial task to connect the function fields with curves over number fields by refactoring the curve framework. 

Having this done well (and efficiently) would be a major step towards bringing Sage closer to Magma's capabilities in this area.

A major gap between Sage and Magma in this area is in performance speed. 

[Kwankyu Lee]

On Saturday, March 28, 2020 at 1:05:01 AM UTC+9, Paddy Adams wrote:

At the moment the implementation of algebraic curves in Sage doesn't seem to be connected at all to the implementation of Function Fields. Is this intentional? It would seem to be useful to be able to, for example, get the function field from a curve. Also surely the Riemann-Roch basis function on projective curves should return elements of a function field and not, as it does currently, the field of fractions of k[X,Y,Z].

Thanks to the current global meltdown I am stuck at home with time on my hands so happy to try and tie together these two areas if there isn't a specific reason not to.

Then try the latest Sage beta and see

https://trac.sagemath.org/ticket/27953  

[John Cremona]

On Sat, 28 Mar 2020, 12:06 Kwankyu Lee, <ekwa...@gmail.com> wrote:

On Saturday, March 28, 2020 at 6:16:25 PM UTC+9, John Cremona wrote:

Certainly this is not intentional. It just has not been done.  

Wrong. It was recently done for curves over finite fields in affine and projective spaces.

Sorry about that -- thanks for the correction.

As we already have function fields over number fields, it would be a rather trivial task to connect the function fields with curves over number fields by refactoring the curve framework. 

I hope you are right about this being easy. From an arithmetic geometry perspective it seems to be a 2-dimensional rather than 1-dimensional situation.

Having this done well (and efficiently) would be a major step towards bringing Sage closer to Magma's capabilities in this area.

A major gap between Sage and Magma in this area is in performance speed. 

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