Categoricity of modular and Shimura curves

Daw, C. ORCID: https://orcid.org/0000-0002-2488-6729 and Harris, A. (2017) Categoricity of modular and Shimura curves. Journal of the Institute of Mathematics of Jussieu. ISSN 1475-3030 doi: 10.1017/S1474748015000365

Abstract/Summary

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 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 L ω 1 ,ω Lω1,ω -sentence having a unique model of cardinality ℵ 1 ℵ1 is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this Lω1,ω -sentence has a unique model in every infinite cardinality. ℵ1 is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this Lω1,ω -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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/70362
Identification Number/DOI	10.1017/S1474748015000365
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Cambridge University Press
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