Modular embeddings and automorphic Higgs bundles

Abstract

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 nonarithmetic) triangle groups by permuting the defining ideals in the tautological way.