Map between projective curves defined in an extension field

asked 2017-07-14 18:08:01 +0200

Road gravatar image

updated 2019-06-20 13:09:08 +0200

FrédéricC gravatar image

For example, suppose I have the following 2 projective curves:

k = GF(13)
x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens()
E = Curve(2*x^2 + 8*y*z + 8*z^2)
W = Curve(x^2 + y*z + z^2)

I like to define a map from E to W that involves $\sqrt 2$ and $\sqrt 8$, which do not exist in k = GF(13), but do in an extension of k:

x = PolynomialRing(k,'x').gen()
K = GF(13**2, 'w', modulus=x^2-2)
w = K.gen()

So $w = \sqrt 2$ and $2w = \sqrt 8$. The map I like to define sends $(x:y:z)$ to $(wx:2wy:2wz)$.

In this particular example, it's obvious that $(wx:2wy:2wz) = (x:2y:2z)$; but it's just a simple example do demonstrate the problem.

Something like this doesn't work:

x,y,z = ProjectiveSpace(k, 2, 'x,y,z').gens() #or ProjectiveSpace(K, 2, 'x,y,z').gens()
E.Hom(W)([w*x, 2*w*y, 2*w*z])

Thank you.

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The following works for me:

sage: K.<a> = GF(13^2) 
sage: x,y,z = PolynomialRing(K, names='x,y,z').gens()
sage: E = Curve(2*x^2 + 8*y*z + 8*z^2)
sage: W = Curve(x^2 + y*z + z^2)
sage: E.Hom(W)([a*x, 2*a*y, 2*a*z])
Scheme morphism:
  From: Projective Conic Curve over Finite Field in a of size 13^2 defined by 2*x^2 - 5*y*z - 5*z^2
  To:   Projective Conic Curve over Finite Field in a of size 13^2 defined by x^2 + y*z + z^2
  Defn: Defined on coordinates by sending (x : y : z) to
        ((a)*x : (2*a)*y : (2*a)*z)

The variables x,y,z are in my case generators of a polynomial ring.

dan_fulea gravatar imagedan_fulea ( 2017-07-14 19:31:10 +0200 )edit