Categoricity of modular and Shimura curves

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.