2015/09/02 by Robert A. Kucharczyk, Kucharczyk, Robert A.
Mathematics · #11F99 #11G18 #11G30 #11G35 #14G30 #14G35 #14H60 #34M45 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11F41 #Secondary: 11F03 #math.AG #math.DG #math.NT #msc:11F03 #msc:11F41 #msc:11F99 #msc:11G18 #msc:11G30 #msc:11G35 #msc:14G30 #msc:14G35 #msc:14H60 #msc:34M45
paper · pdf · doi:10.48550/arxiv.1509.00893
41 pages
arxiv created 2015/09/02 · arxiv updated 2015/09/04
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings and apply our criteria to study the effect of abstract field automorphisms of ℂ on modular embeddings. Finally we derive that the absolute Galois group of ℚ operates on the dessins d'enfants defined by principal congruence subgroups of (arithmetic or non-arithmetic) triangle groups by permuting the defining ideals in the tautological way.