### Elliptic curves in 3-space

Posted on June 10, 2019

Suppose that we have an elliptic curve with Weierstrass coordinates I.e., such that defines an isomorphism of with the curve given by with each and distinguished point Let us from now on identify with the isomorphic Weierstrass plane curve and let be explicit Weierstrass coordinates for

The map given by maps into the curve given by the intersection of and in Observe that this map is bijectve on the affine patch as we then dehomogenize coordinates with and find We also find that there is a single point in satisfying Observe that generate and Therefore is a local parameter at and has a order at Observe that and thus has a pole of order at ; thus has a pole of order Similarly we see that has a pole of order at Therefore

Finally, observe that the map given by is a rational inverse that is regular on the affine patch It remains to check the one point in satisfying At we find, via a similar argument to the one given above for the curve , that is a local parameter for at , has order at , and has order Therefore has a pole of order at and has a pole of order at Thus is regular at and Hence is in fact a regular inverse to and is isomorphic to In particular, defines an isogeny between the elliptic curve with base point and the elliptic curve with base point

Recall from Bezout’s theorem in projective space that if define hypersurfaces in in general position, i.e. then counting points with multiplicity. Therefore if is a hyperplane, the above result tells us that We made the observation above that the hyperplane defined by intersects at the point with multiplicity

Suppose that such that Then In this implies for some Observe that is regular on the affine patch since it has only a single pole at Thus there exists a polynomial which defines a hypersurface intersecting at Homogenizing we get a form such that and restricts to Since has order at and intersects with multiplicity , this implies Thus defines a hyperplane in that intersects at

Conversely, if is a hyperplane defined by a form that intersects at points , then clearly such that the divisor of is Thus

Thus four points add to on if and only if there is a hyperplane in intersecting at their images under The idea for this post comes from Silverman (1, Exercise 3.10).

### References

1. Joseph Silverman, The arithmetic of elliptic curves, 2nd ed., Springer Graduate Texts in Mathematics, 1992