The symmetries of the tetrahedral Kummer surface in the Mathieu group M_24

Understand

We provide a method, based on Nikulin's lattice gluing techniques, which identifies the symplectic automorphisms of Kummer surfaces as permutation groups on 24 elements preserving the Golay code.

• In other words, we explicitly realise these symplectic automorphism groups as subgroups of the Mathieu group M_24.
• The example of the tetrahedral Kummer surface is treated in detail, confirming the existence proofs of Mukai and Kondo, that its group of symplectic automorphisms is a subgroup of one of eleven subgroups of the sporadic group known as Mathieu group M_23.
• Kondo's lattice construction, which uses a different gluing technique from the one advocated here to rederive Mukai's results, is reviewed, and a slight generalisation is used to check the consistency of our results.