# Galois conjugates of special points and special subvarieties in Shimura varieties

## TL;DR

This paper proves that Galois conjugation preserves special points and subvarieties in Shimura varieties, and that conjugates of a special point share the same complexity, based on canonical models and Langlands' conjecture.

## Contribution

It establishes the Galois invariance of special points and subvarieties in Shimura varieties, extending understanding of their arithmetic and geometric properties.

## Key findings

- Galois action maps special points to special points
- Galois conjugates of a special point have identical complexity
- Results rely on canonical models and Langlands' conjecture

## Abstract

Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih's construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.