CATEGORICITY OF MODULAR AND SHIMURA CURVES. (30th September 2015)

• Record Type: Journal Article
• Authors: Daw, Christopher; Harris, Adam
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 16:Number 5(2017)
• Page Start: 1075
• Page End: 1101
• Publication Date: 2015-09-30
• Subjects: Primary 03C35, -- 11G18, -- 03C10, -- 11G05; categoricity, -- modular curves, -- Shimura varieties, -- open image theorems; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748015000365
• Languages: English
• ISSNs: 1474-7480

Abstract

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 ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1}, \unicode[STIX]{x1D714}}$ -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 ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1}, \unicode[STIX]{x1D714}}$ -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.