The Kummer surface K associated to the Jacobian J of a genus 2 curve is the quotient of J by the inverse map. It can be embedded as a quartic hypersurface in projective 3-space whose only singularities are 16 ordinary double points. The Jacobian is a double cover of K ramified at these double points; they are the images of the two-torsion points on J. Resolving the singularities on K yields a K3-surface.
Currently, Kummer surfaces in Magma are not schemes, but are of type SrfKum. They can be used to perform arithmetic on the Jacobian without the need for reduction of divisors. The other nontrivial functionality that uses them is point searching.
The Kummer surface and arithmetic on it are implemented for Jacobians in arbitrary characteristic following [Mül10b] which extends earlier work by Flynn, described in chapter 3 of [CF96].
The Kummer surface of the Jacobian J of a genus 2 curve.
The defining polynomial of the Kummer surface K.
The base field of the Kummer surface K.
Extends the base field of the Kummer surface K to the field F.
Extends the base field of the Kummer surface K by the map j, where j is a ring homomorphism with the base field of C as its domain.
Extends the finite base field of the Kummer surface K over a finite field to the degree n extension.