Categoricity of modular and Shimura curves

TL;DR

This paper links model theory and arithmetic geometry to establish a categoricity result for modular and Shimura curves, showing a unique model in all infinite cardinalities and characterizing special points.

Contribution

It introduces a model-theoretic framework for Shimura varieties and proves a new categoricity result for modular and Shimura curves, connecting Galois representations and special points.

Findings

Unique model of the $orall ext{L}_{ ext{omega}_1, ext{omega}}$-sentence in all infinite cardinalities for modular and Shimura curves.

New characterization of special points on Shimura varieties.

Establishment of a link between model-theoretic categoricity and arithmetic properties of Galois representations.

Abstract

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain $\mathcal{L}_{\omega_1, \omega}$-sentence having a unique model of cardinality $\aleph_1$ is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this $\mathcal{L}_{\omega_1, \omega}$-sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.