Categoricity of Shimura Varieties

TL;DR

This paper develops a model-theoretic framework for Shimura varieties, establishing conditions for categoricity linked to number theory conjectures, and proves categoricity for specific cases of moduli spaces of abelian varieties.

Contribution

It introduces necessary and sufficient model-theoretic conditions for Shimura varieties to be categorical, connecting these to key number theory conjectures.

Findings

Categoricity conditions are linked to Galois representations.

Proves categoricity for f6A_2 and f6A_3.

Existing literature suffices for these proofs.

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 $\mathcal{A}_{3}$.