MAGMA Computational Algebra System

Kummer Surfaces

Currently the only Kummer surfaces in Magma are the Kummer surfaces associated to Jacobians of genus 2 curves. (The Kummer surface associated to a Jacobian J is a desingularisation of the quotient of J by the inverse map.) They are implemented mainly to help with computations on Jacobians. The main nontrivial functionality for Kummer surfaces is point searching.

Subsections

• Creation of a Kummer Surface
• Structure Operations
• Base Ring
• Changing the Base Ring

Creation of a Kummer Surface

KummerSurface(J) : JacHyp -> SrfKum

The Kummer surface of the Jacobian J of a genus 2 curve.

Structure Operations

DefiningPolynomial(K) : SrfKum -> RngMPolElt

The defining polynomial of the Kummer surface K.

Base Ring

BaseField(K) : SrfKum -> Fld

BaseRing(K) : SrfKum -> Rng

CoefficientRing(K) : SrfKum -> Rng

The base field of the Kummer surface K.

Changing the Base Ring

BaseChange(K, F) : SrfKum, Rng -> SrfKum

BaseExtend(K, F) : SrfKum, Rng -> SrfKum

Extends the base field of the Kummer surface K to the field F.

BaseChange(K, j) : SrfKum, Map -> SrfKum

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.

BaseChange(K, n): SrfKum, RngIntElt -> SrfKum

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.