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.

References in corpus (1)

• Heights of pre-special points of Shimura varieties

Cited by in corpus (2)

• Heights of pre-special points of Shimura varieties
• Zilber's notion of logically perfect structure: Universal Covers