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arXiv · 1509.00893

Modular embeddings and automorphic Higgs bundles

Robert A. Kucharczyk
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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.

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Robert A. Kucharczyk. 2015-09-02. Modular embeddings and automorphic Higgs bundles. https://arxiv.org/abs/1509.00893

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