2010/08/05 by Anne Taormina, Taormina, Anne, Katrin Wendland +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #14J28 #14J50 #20B25 #81T40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #FOS: Physical sciences #Group Theory (math.GR) #High Energy Physics - Theory (hep-th) #hep-th #math.AG #math.GR #msc:14J28 #msc:14J50 #msc:20B25 #msc:81T40
paper · pdf · doi:10.48550/arxiv.1008.0954
The paper withdrawn is mathematically correct, but is superseded by arXiv:1107.3834, where crucial new insights on symmetries are included
openalex publication_date 2010/08/05 · arxiv created 2011/07/20 · arxiv updated 2011/07/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 M24. 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 M23. 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. The framework presented here provides a line of attack to unravel the role of the sporadic Mathieu group Mathieu M24, of which M23 is a subgroup of index 24, when searching for symmetries beyond the classical symplectic automorphisms in the context of strings compactified on a K3 surface.