2012/12/05 by Miranda C. N. Cheng, Cheng, Miranda C. N., John F. R. Duncan +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT
paper · pdf · doi:10.48550/arxiv.1212.0906
18 pages, published version; AMS Proceeding of the Conference "Perspectives in Representation Theory", 2013; broken reference links repaired in this version
openalex publication_date 2012/12/05 · arxiv created 2013/08/25 · arxiv updated 2013/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that a certain space of cusp forms for the Hecke congruence group of a given level is one-dimensional if and only if that level is the order of an element of the second largest Mathieu group. As such, our result furnishes a direct analogue of Ogg's observation that the normaliser of a Hecke congruence group of prime level has genus zero if and only if that prime divides the order of the Fischer-Griess monster group. The significance of the cusp forms under consideration is explained by the Rademacher sum construction of the McKay-Thompson series of Mathieu moonshine. Our result supports a conjectural characterisation of the discrete groups and multiplier systems arising in Mathieu moonshine.