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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.
BaseRing(K) : SrfKum -> Rng
CoefficientRing(K) : SrfKum -> Rng
The base field of the Kummer surface K.
BaseExtend(K, F) : SrfKum, Rng -> SrfKum
Extends the base field of the Kummer surface K to
the field F.
BaseExtend(K, j) : SrfKum, Map -> SrfKum
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.
BaseExtend(K, n): SrfKum, RngIntElt -> SrfKum
Extends the finite base field of the Kummer surface K over
a finite field to the degree n extension.
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