Modular embeddings and automorphic Higgs bundles

TL;DR

This paper explores criteria for modular embeddings of complex curves into quaternionic Shimura varieties, develops an adelic framework, and examines the Galois action on dessins d'enfants related to triangle groups.

Contribution

It introduces new algebro-geometric and topological criteria for modular embeddings and analyzes the Galois group's permutation of dessins d'enfants via an adelic formalism.

Findings

Established criteria for modular embeddings in quaternionic Shimura varieties.

Developed an adelic formalism for studying modular embeddings.

Showed the Galois group permutes dessins d'enfants by automorphisms of $\

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 $\mathbb{C}$ on modular embeddings. Finally we derive that the absolute Galois group of $\mathbb{Q}$ 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.