Categoricity of Shimura Varieties

Abstract

We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity. We show that these conditions are directly related to important conjectures in number theory coming from Galois representations attached the points of a Shimura variety. We end by showing that the existing literature is enough to prove categoricity of $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$.